Theorem poimirlem27 33436

Description: Lemma for poimir 33442 showing that the difference between admissible faces in the whole cube and admissible faces on the back face is even. Equation (7) of [Kulpa] p. 548. (Contributed by Brendan Leahy, 21-Aug-2020.)

Hypotheses

Ref	Expression
poimir.0	⊢ (𝜑 → 𝑁 ∈ ℕ)
poimirlem28.1	⊢ (𝑝 = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)
poimirlem28.2	⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))
poimirlem28.3	⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛)
poimirlem28.4	⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1))

Assertion

Ref	Expression
poimirlem27	⊢ (𝜑 → 2 ∥ ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)})))

Distinct variable groups:   𝑓,𝑖,𝑗,𝑛,𝑝,𝑠,𝑡   𝜑,𝑗,𝑛   𝑗,𝑁,𝑛   𝜑,𝑖,𝑝,𝑠,𝑡   𝐵,𝑓,𝑖,𝑗,𝑛,𝑠,𝑡   𝑓,𝐾,𝑖,𝑗,𝑛,𝑝,𝑠,𝑡   𝑓,𝑁,𝑖,𝑝,𝑠,𝑡   𝐶,𝑖,𝑛,𝑝,𝑡

Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑝)   𝐶(𝑓,𝑗,𝑠)

Proof of Theorem poimirlem27

Dummy variables 𝑚 𝑞 𝑢 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfi 12771	. . . . . 6 ⊢ (0...𝐾) ∈ Fin
2		fzfi 12771	. . . . . 6 ⊢ (1...𝑁) ∈ Fin
3		mapfi 8262	. . . . . 6 ⊢ (((0...𝐾) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((0...𝐾) ↑𝑚 (1...𝑁)) ∈ Fin)
4	1, 2, 3	mp2an 708	. . . . 5 ⊢ ((0...𝐾) ↑𝑚 (1...𝑁)) ∈ Fin
5		fzfi 12771	. . . . 5 ⊢ (0...(𝑁 − 1)) ∈ Fin
6		mapfi 8262	. . . . 5 ⊢ ((((0...𝐾) ↑𝑚 (1...𝑁)) ∈ Fin ∧ (0...(𝑁 − 1)) ∈ Fin) → (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1))) ∈ Fin)
7	4, 5, 6	mp2an 708	. . . 4 ⊢ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1))) ∈ Fin
8	7	a1i 11	. . 3 ⊢ (𝜑 → (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1))) ∈ Fin)
9		2z 11409	. . . 4 ⊢ 2 ∈ ℤ
10	9	a1i 11	. . 3 ⊢ (𝜑 → 2 ∈ ℤ)
11		fzofi 12773	. . . . . . . 8 ⊢ (0..^𝐾) ∈ Fin
12		mapfi 8262	. . . . . . . 8 ⊢ (((0..^𝐾) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((0..^𝐾) ↑𝑚 (1...𝑁)) ∈ Fin)
13	11, 2, 12	mp2an 708	. . . . . . 7 ⊢ ((0..^𝐾) ↑𝑚 (1...𝑁)) ∈ Fin
14		mapfi 8262	. . . . . . . . 9 ⊢ (((1...𝑁) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((1...𝑁) ↑𝑚 (1...𝑁)) ∈ Fin)
15	2, 2, 14	mp2an 708	. . . . . . . 8 ⊢ ((1...𝑁) ↑𝑚 (1...𝑁)) ∈ Fin
16		f1of 6137	. . . . . . . . . 10 ⊢ (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑓:(1...𝑁)⟶(1...𝑁))
17	16	ss2abi 3674	. . . . . . . . 9 ⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)}
18		ovex 6678	. . . . . . . . . 10 ⊢ (1...𝑁) ∈ V
19	18, 18	mapval 7869	. . . . . . . . 9 ⊢ ((1...𝑁) ↑𝑚 (1...𝑁)) = {𝑓 ∣ 𝑓:(1...𝑁)⟶(1...𝑁)}
20	17, 19	sseqtr4i 3638	. . . . . . . 8 ⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ ((1...𝑁) ↑𝑚 (1...𝑁))
21		ssfi 8180	. . . . . . . 8 ⊢ ((((1...𝑁) ↑𝑚 (1...𝑁)) ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ⊆ ((1...𝑁) ↑𝑚 (1...𝑁))) → {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin)
22	15, 20, 21	mp2an 708	. . . . . . 7 ⊢ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin
23		xpfi 8231	. . . . . . 7 ⊢ ((((0..^𝐾) ↑𝑚 (1...𝑁)) ∈ Fin ∧ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ∈ Fin) → (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin)
24	13, 22, 23	mp2an 708	. . . . . 6 ⊢ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin
25		fzfi 12771	. . . . . 6 ⊢ (0...𝑁) ∈ Fin
26		xpfi 8231	. . . . . 6 ⊢ (((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin ∧ (0...𝑁) ∈ Fin) → ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin)
27	24, 25, 26	mp2an 708	. . . . 5 ⊢ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin
28		rabfi 8185	. . . . 5 ⊢ (((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ∈ Fin)
29	27, 28	ax-mp 5	. . . 4 ⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ∈ Fin
30		hashcl 13147	. . . . 5 ⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ∈ Fin → (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) ∈ ℕ0)
31	30	nn0zd 11480	. . . 4 ⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ∈ Fin → (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) ∈ ℤ)
32	29, 31	mp1i 13	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) → (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) ∈ ℤ)
33		dfrex2 2996	. . . . 5 ⊢ (∃𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ↔ ¬ ∀𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))
34		nfv 1843	. . . . . 6 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1))))
35		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑡2
36		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑡 ∥
37		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑡#
38		nfrab1 3122	. . . . . . . 8 ⊢ Ⅎ𝑡{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}
39	37, 38	nffv 6198	. . . . . . 7 ⊢ Ⅎ𝑡(#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})
40	35, 36, 39	nfbr 4699	. . . . . 6 ⊢ Ⅎ𝑡2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})
41		neq0 3930	. . . . . . . . . . . 12 ⊢ (¬ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = ∅ ↔ ∃𝑠 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})
42		iddvds 14995	. . . . . . . . . . . . . . . . 17 ⊢ (2 ∈ ℤ → 2 ∥ 2)
43	9, 42	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ 2 ∥ 2
44		vex 3203	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑠 ∈ V
45		hashsng 13159	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ V → (#‘{𝑠}) = 1)
46	44, 45	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (#‘{𝑠}) = 1
47	46	oveq2i 6661	. . . . . . . . . . . . . . . . 17 ⊢ (1 + (#‘{𝑠})) = (1 + 1)
48		df-2 11079	. . . . . . . . . . . . . . . . 17 ⊢ 2 = (1 + 1)
49	47, 48	eqtr4i 2647	. . . . . . . . . . . . . . . 16 ⊢ (1 + (#‘{𝑠})) = 2
50	43, 49	breqtrri 4680	. . . . . . . . . . . . . . 15 ⊢ 2 ∥ (1 + (#‘{𝑠}))
51		rabfi 8185	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∈ Fin)
52		diffi 8192	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∈ Fin → ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ Fin)
53	27, 51, 52	mp2b 10	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ Fin
54		snfi 8038	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑠} ∈ Fin
55		incom 3805	. . . . . . . . . . . . . . . . . . . 20 ⊢ (({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∩ {𝑠}) = ({𝑠} ∩ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}))
56		disjdif 4040	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({𝑠} ∩ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = ∅
57	55, 56	eqtri 2644	. . . . . . . . . . . . . . . . . . 19 ⊢ (({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∩ {𝑠}) = ∅
58		hashun 13171	. . . . . . . . . . . . . . . . . . 19 ⊢ ((({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ Fin ∧ {𝑠} ∈ Fin ∧ (({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∩ {𝑠}) = ∅) → (#‘(({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∪ {𝑠})) = ((#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (#‘{𝑠})))
59	53, 54, 57, 58	mp3an 1424	. . . . . . . . . . . . . . . . . 18 ⊢ (#‘(({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∪ {𝑠})) = ((#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (#‘{𝑠}))
60		difsnid 4341	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → (({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∪ {𝑠}) = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})
61	60	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → (#‘(({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∪ {𝑠})) = (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))
62	59, 61	syl5eqr 2670	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → ((#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (#‘{𝑠})) = (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))
63	62	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → ((#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (#‘{𝑠})) = (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))
64		poimir.0	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ ℕ)
65	64	ad3antrrr 766	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → 𝑁 ∈ ℕ)
66		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑢 → (2nd ‘𝑡) = (2nd ‘𝑢))
67	66	breq2d 4665	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑢 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑢)))
68	67	ifbid 4108	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑢 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑢), 𝑦, (𝑦 + 1)))
69	68	csbeq1d 3540	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑢 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd ‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
70		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑢 → (1st ‘𝑡) = (1st ‘𝑢))
71	70	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑢 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑢)))
72	70	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑡 = 𝑢 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑢)))
73	72	imaeq1d 5465	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑡 = 𝑢 → ((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑢)) “ (1...𝑗)))
74	73	xpeq1d 5138	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑢 → (((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}))
75	72	imaeq1d 5465	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑡 = 𝑢 → ((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)))
76	75	xpeq1d 5138	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑢 → (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))
77	74, 76	uneq12d 3768	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑢 → ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))
78	71, 77	oveq12d 6668	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑢 → ((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st ‘𝑢)) ∘𝑓 + ((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))))
79	78	csbeq2dv 3992	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑢 → ⦋if(𝑦 < (2nd ‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd ‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑢)) ∘𝑓 + ((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))))
80	69, 79	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑢 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd ‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑢)) ∘𝑓 + ((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))))
81	80	mpteq2dv 4745	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑢 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑢)) ∘𝑓 + ((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))))
82		breq1 4656	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 = 𝑤 → (𝑦 < (2nd ‘𝑢) ↔ 𝑤 < (2nd ‘𝑢)))
83		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 = 𝑤 → 𝑦 = 𝑤)
84		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 = 𝑤 → (𝑦 + 1) = (𝑤 + 1))
85	82, 83, 84	ifbieq12d 4113	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = 𝑤 → if(𝑦 < (2nd ‘𝑢), 𝑦, (𝑦 + 1)) = if(𝑤 < (2nd ‘𝑢), 𝑤, (𝑤 + 1)))
86	85	csbeq1d 3540	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑤 → ⦋if(𝑦 < (2nd ‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑢)) ∘𝑓 + ((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑤 < (2nd ‘𝑢), 𝑤, (𝑤 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑢)) ∘𝑓 + ((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))))
87		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 = 𝑖 → (1...𝑗) = (1...𝑖))
88	87	imaeq2d 5466	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 = 𝑖 → ((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑢)) “ (1...𝑖)))
89	88	xpeq1d 5138	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 = 𝑖 → (((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑢)) “ (1...𝑖)) × {1}))
90		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 = 𝑖 → (𝑗 + 1) = (𝑖 + 1))
91	90	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 = 𝑖 → ((𝑗 + 1)...𝑁) = ((𝑖 + 1)...𝑁))
92	91	imaeq2d 5466	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 = 𝑖 → ((2nd ‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)))
93	92	xpeq1d 5138	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 = 𝑖 → (((2nd ‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))
94	89, 93	uneq12d 3768	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 = 𝑖 → ((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd ‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})))
95	94	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 𝑖 → ((1st ‘(1st ‘𝑢)) ∘𝑓 + ((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st ‘𝑢)) ∘𝑓 + ((((2nd ‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd ‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))))
96	95	cbvcsbv 3539	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ⦋if(𝑤 < (2nd ‘𝑢), 𝑤, (𝑤 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑢)) ∘𝑓 + ((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑤 < (2nd ‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1st ‘(1st ‘𝑢)) ∘𝑓 + ((((2nd ‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd ‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})))
97	86, 96	syl6eq 2672	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑤 → ⦋if(𝑦 < (2nd ‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑢)) ∘𝑓 + ((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑤 < (2nd ‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1st ‘(1st ‘𝑢)) ∘𝑓 + ((((2nd ‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd ‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))))
98	97	cbvmptv 4750	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑢), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑢)) ∘𝑓 + ((((2nd ‘(1st ‘𝑢)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑢)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑤 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑤 < (2nd ‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1st ‘(1st ‘𝑢)) ∘𝑓 + ((((2nd ‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd ‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))))
99	81, 98	syl6eq 2672	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑢 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑤 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑤 < (2nd ‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1st ‘(1st ‘𝑢)) ∘𝑓 + ((((2nd ‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd ‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0})))))
100	99	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑢 → (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝑥 = (𝑤 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑤 < (2nd ‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1st ‘(1st ‘𝑢)) ∘𝑓 + ((((2nd ‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd ‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))))))
101	100	cbvrabv 3199	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = {𝑢 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑤 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑤 < (2nd ‘𝑢), 𝑤, (𝑤 + 1)) / 𝑖⦌((1st ‘(1st ‘𝑢)) ∘𝑓 + ((((2nd ‘(1st ‘𝑢)) “ (1...𝑖)) × {1}) ∪ (((2nd ‘(1st ‘𝑢)) “ ((𝑖 + 1)...𝑁)) × {0}))))}
102		elmapi 7879	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1))) → 𝑥:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))
103	102	ad3antlr 767	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → 𝑥:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))
104		simpr 477	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})
105		simpl 473	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) → ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)
106	105	ralimi 2952	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)
107	106	ad2antlr 763	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)
108		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑚 → (𝑝‘𝑛) = (𝑝‘𝑚))
109	108	neeq1d 2853	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑚 → ((𝑝‘𝑛) ≠ 0 ↔ (𝑝‘𝑚) ≠ 0))
110	109	rexbidv 3052	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑚) ≠ 0))
111		fveq1 6190	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑝 = 𝑞 → (𝑝‘𝑚) = (𝑞‘𝑚))
112	111	neeq1d 2853	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 = 𝑞 → ((𝑝‘𝑚) ≠ 0 ↔ (𝑞‘𝑚) ≠ 0))
113	112	cbvrexv 3172	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃𝑝 ∈ ran 𝑥(𝑝‘𝑚) ≠ 0 ↔ ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 0)
114	110, 113	syl6bb 276	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 0))
115	114	rspccva 3308	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 0)
116	107, 115	sylan 488	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 0)
117		simpr 477	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) → ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)
118	117	ralimi 2952	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)
119	118	ad2antlr 763	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)
120	108	neeq1d 2853	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑚 → ((𝑝‘𝑛) ≠ 𝐾 ↔ (𝑝‘𝑚) ≠ 𝐾))
121	120	rexbidv 3052	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾 ↔ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑚) ≠ 𝐾))
122	111	neeq1d 2853	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 = 𝑞 → ((𝑝‘𝑚) ≠ 𝐾 ↔ (𝑞‘𝑚) ≠ 𝐾))
123	122	cbvrexv 3172	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃𝑝 ∈ ran 𝑥(𝑝‘𝑚) ≠ 𝐾 ↔ ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 𝐾)
124	121, 123	syl6bb 276	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑚 → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾 ↔ ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 𝐾))
125	124	rspccva 3308	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾 ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 𝐾)
126	119, 125	sylan 488	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) ∧ 𝑚 ∈ (1...𝑁)) → ∃𝑞 ∈ ran 𝑥(𝑞‘𝑚) ≠ 𝐾)
127	65, 101, 103, 104, 116, 126	poimirlem22 33431	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → ∃!𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}𝑧 ≠ 𝑠)
128		eldifsn 4317	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ↔ (𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∧ 𝑧 ≠ 𝑠))
129	128	eubii 2492	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃!𝑧 𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ↔ ∃!𝑧(𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∧ 𝑧 ≠ 𝑠))
130	53	elexi 3213	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ V
131		euhash1 13208	. . . . . . . . . . . . . . . . . . . 20 ⊢ (({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}) ∈ V → ((#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = 1 ↔ ∃!𝑧 𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})))
132	130, 131	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ ((#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = 1 ↔ ∃!𝑧 𝑧 ∈ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠}))
133		df-reu 2919	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃!𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}𝑧 ≠ 𝑠 ↔ ∃!𝑧(𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∧ 𝑧 ≠ 𝑠))
134	129, 132, 133	3bitr4ri 293	. . . . . . . . . . . . . . . . . 18 ⊢ (∃!𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}𝑧 ≠ 𝑠 ↔ (#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = 1)
135	127, 134	sylib 208	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → (#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) = 1)
136	135	oveq1d 6665	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → ((#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ∖ {𝑠})) + (#‘{𝑠})) = (1 + (#‘{𝑠})))
137	63, 136	eqtr3d 2658	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) = (1 + (#‘{𝑠})))
138	50, 137	syl5breqr 4691	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) ∧ 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))
139	138	ex 450	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})))
140	139	exlimdv 1861	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (∃𝑠 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})))
141	41, 140	syl5bi 232	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (¬ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = ∅ → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})))
142		dvds0 14997	. . . . . . . . . . . . . 14 ⊢ (2 ∈ ℤ → 2 ∥ 0)
143	9, 142	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 2 ∥ 0
144		hash0 13158	. . . . . . . . . . . . 13 ⊢ (#‘∅) = 0
145	143, 144	breqtrri 4680	. . . . . . . . . . . 12 ⊢ 2 ∥ (#‘∅)
146		fveq2 6191	. . . . . . . . . . . 12 ⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = ∅ → (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) = (#‘∅))
147	145, 146	syl5breqr 4691	. . . . . . . . . . 11 ⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = ∅ → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))
148	141, 147	pm2.61d2 172	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}))
149	148	ex 450	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) → (∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})))
150	149	adantld 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) → (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})))
151		iba 524	. . . . . . . . . . 11 ⊢ (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))))
152	151	rabbidv 3189	. . . . . . . . . 10 ⊢ (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})
153	152	fveq2d 6195	. . . . . . . . 9 ⊢ (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) = (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}))
154	153	breq2d 4665	. . . . . . . 8 ⊢ (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}) ↔ 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})))
155	150, 154	mpbidi 231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) → (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})))
156	155	a1d 25	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) → (𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}))))
157	34, 40, 156	rexlimd 3026	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) → (∃𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})))
158	33, 157	syl5bir 233	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) → (¬ ∀𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})))
159		simpr 477	. . . . . . . . 9 ⊢ ((𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) → ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))
160	159	con3i 150	. . . . . . . 8 ⊢ (¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → ¬ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))))
161	160	ralimi 2952	. . . . . . 7 ⊢ (∀𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → ∀𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))))
162		rabeq0 3957	. . . . . . 7 ⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} = ∅ ↔ ∀𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))))
163	161, 162	sylibr 224	. . . . . 6 ⊢ (∀𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} = ∅)
164	163	fveq2d 6195	. . . . 5 ⊢ (∀𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) = (#‘∅))
165	145, 164	syl5breqr 4691	. . . 4 ⊢ (∀𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ¬ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}))
166	158, 165	pm2.61d2 172	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) → 2 ∥ (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}))
167	8, 10, 32, 166	fsumdvds 15030	. 2 ⊢ (𝜑 → 2 ∥ Σ𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))(#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}))
168		rabfi 8185	. . . . 5 ⊢ (((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∈ Fin)
169	27, 168	ax-mp 5	. . . 4 ⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∈ Fin
170		simp1 1061	. . . . . . 7 ⊢ ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)
171		sneq 4187	. . . . . . . . . . . . 13 ⊢ ((2nd ‘𝑡) = 𝑁 → {(2nd ‘𝑡)} = {𝑁})
172	171	difeq2d 3728	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑡) = 𝑁 → ((0...𝑁) ∖ {(2nd ‘𝑡)}) = ((0...𝑁) ∖ {𝑁}))
173		difun2 4048	. . . . . . . . . . . . 13 ⊢ (((0...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁}) = ((0...(𝑁 − 1)) ∖ {𝑁})
174	64	nnnn0d 11351	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
175		nn0uz 11722	. . . . . . . . . . . . . . . . . 18 ⊢ ℕ0 = (ℤ≥‘0)
176	174, 175	syl6eleq 2711	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
177		fzm1 12420	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (ℤ≥‘0) → (𝑛 ∈ (0...𝑁) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁)))
178	176, 177	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑛 ∈ (0...𝑁) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁)))
179		elun 3753	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}))
180		velsn 4193	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ {𝑁} ↔ 𝑛 = 𝑁)
181	180	orbi2i 541	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁))
182	179, 181	bitri 264	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (0...(𝑁 − 1)) ∨ 𝑛 = 𝑁))
183	178, 182	syl6bbr 278	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑛 ∈ (0...𝑁) ↔ 𝑛 ∈ ((0...(𝑁 − 1)) ∪ {𝑁})))
184	183	eqrdv 2620	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0...𝑁) = ((0...(𝑁 − 1)) ∪ {𝑁}))
185	184	difeq1d 3727	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((0...𝑁) ∖ {𝑁}) = (((0...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁}))
186	64	nnzd 11481	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℤ)
187		uzid 11702	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁))
188		uznfz 12423	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ≥‘𝑁) → ¬ 𝑁 ∈ (0...(𝑁 − 1)))
189	186, 187, 188	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ 𝑁 ∈ (0...(𝑁 − 1)))
190		disjsn 4246	. . . . . . . . . . . . . . 15 ⊢ (((0...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (0...(𝑁 − 1)))
191		disj3 4021	. . . . . . . . . . . . . . 15 ⊢ (((0...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ (0...(𝑁 − 1)) = ((0...(𝑁 − 1)) ∖ {𝑁}))
192	190, 191	bitr3i 266	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑁 ∈ (0...(𝑁 − 1)) ↔ (0...(𝑁 − 1)) = ((0...(𝑁 − 1)) ∖ {𝑁}))
193	189, 192	sylib 208	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0...(𝑁 − 1)) = ((0...(𝑁 − 1)) ∖ {𝑁}))
194	173, 185, 193	3eqtr4a 2682	. . . . . . . . . . . 12 ⊢ (𝜑 → ((0...𝑁) ∖ {𝑁}) = (0...(𝑁 − 1)))
195	172, 194	sylan9eqr 2678	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (2nd ‘𝑡) = 𝑁) → ((0...𝑁) ∖ {(2nd ‘𝑡)}) = (0...(𝑁 − 1)))
196	195	rexeqdv 3145	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (2nd ‘𝑡) = 𝑁) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
197	196	biimprd 238	. . . . . . . . 9 ⊢ ((𝜑 ∧ (2nd ‘𝑡) = 𝑁) → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
198	197	ralimdv 2963	. . . . . . . 8 ⊢ ((𝜑 ∧ (2nd ‘𝑡) = 𝑁) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
199	198	expimpd 629	. . . . . . 7 ⊢ (𝜑 → (((2nd ‘𝑡) = 𝑁 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
200	170, 199	sylan2i 687	. . . . . 6 ⊢ (𝜑 → (((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁)) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
201	200	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁)) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
202	201	ss2rabdv 3683	. . . 4 ⊢ (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶})
203		hashssdif 13200	. . . 4 ⊢ (({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∈ Fin ∧ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) → (#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))})) = ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) − (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))})))
204	169, 202, 203	sylancr 695	. . 3 ⊢ (𝜑 → (#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))})) = ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) − (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))})))
205	64	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → 𝑁 ∈ ℕ)
206		poimirlem28.1	. . . . . . . . . 10 ⊢ (𝑝 = ((1st ‘𝑠) ∘𝑓 + ((((2nd ‘𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd ‘𝑠) “ ((𝑗 + 1)...𝑁)) × {0}))) → 𝐵 = 𝐶)
207		poimirlem28.2	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))
208	207	adantlr 751	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → 𝐵 ∈ (0...𝑁))
209		xp1st 7198	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
210		xp1st 7198	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st ‘𝑡)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
211		elmapi 7879	. . . . . . . . . . . 12 ⊢ ((1st ‘(1st ‘𝑡)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st ‘𝑡)):(1...𝑁)⟶(0..^𝐾))
212	209, 210, 211	3syl 18	. . . . . . . . . . 11 ⊢ (𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘(1st ‘𝑡)):(1...𝑁)⟶(0..^𝐾))
213	212	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (1st ‘(1st ‘𝑡)):(1...𝑁)⟶(0..^𝐾))
214		xp2nd 7199	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st ‘𝑡)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
215		fvex 6201	. . . . . . . . . . . . . 14 ⊢ (2nd ‘(1st ‘𝑡)) ∈ V
216		f1oeq1 6127	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (2nd ‘(1st ‘𝑡)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st ‘𝑡)):(1...𝑁)–1-1-onto→(1...𝑁)))
217	215, 216	elab 3350	. . . . . . . . . . . . 13 ⊢ ((2nd ‘(1st ‘𝑡)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st ‘𝑡)):(1...𝑁)–1-1-onto→(1...𝑁))
218	214, 217	sylib 208	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st ‘𝑡)):(1...𝑁)–1-1-onto→(1...𝑁))
219	209, 218	syl 17	. . . . . . . . . . 11 ⊢ (𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd ‘(1st ‘𝑡)):(1...𝑁)–1-1-onto→(1...𝑁))
220	219	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (2nd ‘(1st ‘𝑡)):(1...𝑁)–1-1-onto→(1...𝑁))
221		xp2nd 7199	. . . . . . . . . . 11 ⊢ (𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd ‘𝑡) ∈ (0...𝑁))
222	221	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (2nd ‘𝑡) ∈ (0...𝑁))
223	205, 206, 208, 213, 220, 222	poimirlem24 33433	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋⟨(1st ‘(1st ‘𝑡)), (2nd ‘(1st ‘𝑡))⟩ / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁)))))
224	209	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
225		1st2nd2 7205	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘𝑡) = ⟨(1st ‘(1st ‘𝑡)), (2nd ‘(1st ‘𝑡))⟩)
226	225	csbeq1d 3540	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋⟨(1st ‘(1st ‘𝑡)), (2nd ‘(1st ‘𝑡))⟩ / 𝑠⦌𝐶)
227	226	eqeq2d 2632	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋⟨(1st ‘(1st ‘𝑡)), (2nd ‘(1st ‘𝑡))⟩ / 𝑠⦌𝐶))
228	227	rexbidv 3052	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋⟨(1st ‘(1st ‘𝑡)), (2nd ‘(1st ‘𝑡))⟩ / 𝑠⦌𝐶))
229	228	ralbidv 2986	. . . . . . . . . . 11 ⊢ ((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋⟨(1st ‘(1st ‘𝑡)), (2nd ‘(1st ‘𝑡))⟩ / 𝑠⦌𝐶))
230	229	anbi1d 741	. . . . . . . . . 10 ⊢ ((1st ‘𝑡) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋⟨(1st ‘(1st ‘𝑡)), (2nd ‘(1st ‘𝑡))⟩ / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁)))))
231	224, 230	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋⟨(1st ‘(1st ‘𝑡)), (2nd ‘(1st ‘𝑡))⟩ / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁)))))
232	223, 231	bitr4d 271	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁)))))
233		frn 6053	. . . . . . . . . . . . . . 15 ⊢ (𝑥:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)) → ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁)))
234	102, 233	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1))) → ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁)))
235	234	anim2i 593	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) → (𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))))
236		dfss3 3592	. . . . . . . . . . . . . 14 ⊢ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ↔ ∀𝑛 ∈ (0...(𝑁 − 1))𝑛 ∈ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵))
237		vex 3203	. . . . . . . . . . . . . . . 16 ⊢ 𝑛 ∈ V
238		eqid 2622	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ ran 𝑥 ↦ 𝐵) = (𝑝 ∈ ran 𝑥 ↦ 𝐵)
239	238	elrnmpt 5372	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ V → (𝑛 ∈ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ↔ ∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵))
240	237, 239	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ↔ ∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)
241	240	ralbii 2980	. . . . . . . . . . . . . 14 ⊢ (∀𝑛 ∈ (0...(𝑁 − 1))𝑛 ∈ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ↔ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)
242	236, 241	sylbb 209	. . . . . . . . . . . . 13 ⊢ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) → ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)
243		1eluzge0 11732	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ (ℤ≥‘0)
244		fzss1 12380	. . . . . . . . . . . . . . . . 17 ⊢ (1 ∈ (ℤ≥‘0) → (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1)))
245		ssralv 3666	. . . . . . . . . . . . . . . . 17 ⊢ ((1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1)) → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵))
246	243, 244, 245	mp2b 10	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵)
247	64	nncnd 11036	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑁 ∈ ℂ)
248		npcan1 10455	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
249	247, 248	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
250		peano2zm 11420	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
251	186, 250	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
252		uzid 11702	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ≥‘(𝑁 − 1)))
253		peano2uz 11741	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑁 − 1)))
254	251, 252, 253	3syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑁 − 1)))
255	249, 254	eqeltrrd 2702	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑁 − 1)))
256		fzss2 12381	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ (ℤ≥‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
257	255, 256	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
258	257	sselda 3603	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...𝑁))
259	258	adantlr 751	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...𝑁))
260		simplr 792	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁)))
261		ssel2 3598	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝 ∈ ((0...𝐾) ↑𝑚 (1...𝑁)))
262		elmapi 7879	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 ∈ ((0...𝐾) ↑𝑚 (1...𝑁)) → 𝑝:(1...𝑁)⟶(0...𝐾))
263	261, 262	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝:(1...𝑁)⟶(0...𝐾))
264	260, 263	sylan 488	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → 𝑝:(1...𝑁)⟶(0...𝐾))
265		poimirlem28.3	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝐵 < 𝑛)
266		elfzelz 12342	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℤ)
267	266	zred 11482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑛 ∈ (1...𝑁) → 𝑛 ∈ ℝ)
268	267	ltnrd 10171	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑛 ∈ (1...𝑁) → ¬ 𝑛 < 𝑛)
269		breq1 4656	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑛 = 𝐵 → (𝑛 < 𝑛 ↔ 𝐵 < 𝑛))
270	269	notbid 308	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑛 = 𝐵 → (¬ 𝑛 < 𝑛 ↔ ¬ 𝐵 < 𝑛))
271	268, 270	syl5ibcom 235	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 ∈ (1...𝑁) → (𝑛 = 𝐵 → ¬ 𝐵 < 𝑛))
272	271	necon2ad 2809	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 ∈ (1...𝑁) → (𝐵 < 𝑛 → 𝑛 ≠ 𝐵))
273	272	3ad2ant1 1082	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0) → (𝐵 < 𝑛 → 𝑛 ≠ 𝐵))
274	273	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → (𝐵 < 𝑛 → 𝑛 ≠ 𝐵))
275	265, 274	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 0)) → 𝑛 ≠ 𝐵)
276	275	3exp2 1285	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑝:(1...𝑁)⟶(0...𝐾) → ((𝑝‘𝑛) = 0 → 𝑛 ≠ 𝐵))))
277	276	imp31 448	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑝‘𝑛) = 0 → 𝑛 ≠ 𝐵))
278	277	necon2d 2817	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → (𝑛 = 𝐵 → (𝑝‘𝑛) ≠ 0))
279	278	adantllr 755	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → (𝑛 = 𝐵 → (𝑝‘𝑛) ≠ 0))
280	264, 279	syldan 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → (𝑛 = 𝐵 → (𝑝‘𝑛) ≠ 0))
281	280	reximdva 3017	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0))
282	259, 281	syldan 487	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0))
283	282	ralimdva 2962	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) → (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0))
284	283	imp 445	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)
285	246, 284	sylan2 491	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)
286	285	biantrurd 529	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)))
287		nnuz 11723	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ℕ = (ℤ≥‘1)
288	64, 287	syl6eleq 2711	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘1))
289		fzm1 12420	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ (ℤ≥‘1) → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁)))
290	288, 289	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁)))
291		elun 3753	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}))
292	180	orbi2i 541	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 ∈ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁))
293	291, 292	bitri 264	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (1...(𝑁 − 1)) ∨ 𝑛 = 𝑁))
294	290, 293	syl6bbr 278	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↔ 𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁})))
295	294	eqrdv 2620	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
296	295	raleqdv 3144	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∀𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁})∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0))
297		ralunb 3794	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑛 ∈ ((1...(𝑁 − 1)) ∪ {𝑁})∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0))
298	296, 297	syl6bb 276	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0)))
299		fveq2 6191	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑁 → (𝑝‘𝑛) = (𝑝‘𝑁))
300	299	neeq1d 2853	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑁 → ((𝑝‘𝑛) ≠ 0 ↔ (𝑝‘𝑁) ≠ 0))
301	300	rexbidv 3052	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑁 → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0))
302	301	ralsng 4218	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → (∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0))
303	64, 302	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0))
304	303	anbi2d 740	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∀𝑛 ∈ {𝑁}∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0) ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)))
305	298, 304	bitrd 268	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)))
306	305	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...(𝑁 − 1))∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)))
307		0z 11388	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 0 ∈ ℤ
308		1z 11407	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 1 ∈ ℤ
309		fzshftral 12428	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((0 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ ∧ 1 ∈ ℤ) → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ ((0 + 1)...((𝑁 − 1) + 1))[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵))
310	307, 308, 309	mp3an13 1415	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 − 1) ∈ ℤ → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ ((0 + 1)...((𝑁 − 1) + 1))[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵))
311	186, 250, 310	3syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ ((0 + 1)...((𝑁 − 1) + 1))[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵))
312		0p1e1 11132	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (0 + 1) = 1
313	312	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (0 + 1) = 1)
314	313, 249	oveq12d 6668	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ((0 + 1)...((𝑁 − 1) + 1)) = (1...𝑁))
315	314	raleqdv 3144	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (∀𝑚 ∈ ((0 + 1)...((𝑁 − 1) + 1))[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ (1...𝑁)[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵))
316	311, 315	bitrd 268	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ (1...𝑁)[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵))
317		ovex 6678	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 − 1) ∈ V
318		eqeq1 2626	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = (𝑚 − 1) → (𝑛 = 𝐵 ↔ (𝑚 − 1) = 𝐵))
319	318	rexbidv 3052	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = (𝑚 − 1) → (∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵))
320	317, 319	sbcie 3470	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ([(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵)
321	320	ralbii 2980	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑚 ∈ (1...𝑁)[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑚 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵)
322		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 = 𝑛 → (𝑚 − 1) = (𝑛 − 1))
323	322	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑛 → ((𝑚 − 1) = 𝐵 ↔ (𝑛 − 1) = 𝐵))
324	323	rexbidv 3052	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑛 → (∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵 ↔ ∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵))
325	324	cbvralv 3171	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑚 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑚 − 1) = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)
326	321, 325	bitri 264	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑚 ∈ (1...𝑁)[(𝑚 − 1) / 𝑛]∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)
327	316, 326	syl6bb 276	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵 ↔ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵))
328	327	biimpa 501	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)
329	328	adantlr 751	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵)
330		poimirlem28.4	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → 𝐵 ≠ (𝑛 − 1))
331	330	necomd 2849	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...𝑁) ∧ 𝑝:(1...𝑁)⟶(0...𝐾) ∧ (𝑝‘𝑛) = 𝐾)) → (𝑛 − 1) ≠ 𝐵)
332	331	3exp2 1285	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) → (𝑝:(1...𝑁)⟶(0...𝐾) → ((𝑝‘𝑛) = 𝐾 → (𝑛 − 1) ≠ 𝐵))))
333	332	imp31 448	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑝‘𝑛) = 𝐾 → (𝑛 − 1) ≠ 𝐵))
334	333	necon2d 2817	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑛 − 1) = 𝐵 → (𝑝‘𝑛) ≠ 𝐾))
335	334	adantllr 755	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝:(1...𝑁)⟶(0...𝐾)) → ((𝑛 − 1) = 𝐵 → (𝑝‘𝑛) ≠ 𝐾))
336	264, 335	syldan 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑝 ∈ ran 𝑥) → ((𝑛 − 1) = 𝐵 → (𝑝‘𝑛) ≠ 𝐾))
337	336	reximdva 3017	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ 𝑛 ∈ (1...𝑁)) → (∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵 → ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))
338	337	ralimdva 2962	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵 → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))
339	338	imp 445	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑛 − 1) = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)
340	329, 339	syldan 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)
341	340	biantrud 528	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))
342		r19.26 3064	. . . . . . . . . . . . . . 15 ⊢ (∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾) ↔ (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))
343	341, 342	syl6bbr 278	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∀𝑛 ∈ (1...𝑁)∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ↔ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))
344	286, 306, 343	3bitr2d 296	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ran 𝑥 ⊆ ((0...𝐾) ↑𝑚 (1...𝑁))) ∧ ∀𝑛 ∈ (0...(𝑁 − 1))∃𝑝 ∈ ran 𝑥 𝑛 = 𝐵) → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0 ↔ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))
345	235, 242, 344	syl2an 494	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) ∧ (0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵)) → (∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0 ↔ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))
346	345	pm5.32da 673	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) → (((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0) ↔ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))))
347	346	anbi2d 740	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) → ((𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))))
348	347	rexbidva 3049	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ ∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))))
349	348	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑁) ≠ 0)) ↔ ∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))))
350	194	rexeqdv 3145	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
351	350	biimpd 219	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
352	351	ralimdv 2963	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
353	172	rexeqdv 3145	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘𝑡) = 𝑁 → (∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
354	353	ralbidv 2986	. . . . . . . . . . . . . . . . . 18 ⊢ ((2nd ‘𝑡) = 𝑁 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
355	354	imbi1d 331	. . . . . . . . . . . . . . . . 17 ⊢ ((2nd ‘𝑡) = 𝑁 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {𝑁})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)))
356	352, 355	syl5ibrcom 237	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2nd ‘𝑡) = 𝑁 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)))
357	356	com23 86	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 → ((2nd ‘𝑡) = 𝑁 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶)))
358	357	imp 445	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → ((2nd ‘𝑡) = 𝑁 → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
359	358	adantrd 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (((2nd ‘𝑡) = 𝑁 ∧ (((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁)) → ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶))
360	359	pm4.71rd 667	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (((2nd ‘𝑡) = 𝑁 ∧ (((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁)) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑡) = 𝑁 ∧ (((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁)))))
361		an12 838	. . . . . . . . . . . . 13 ⊢ ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑡) = 𝑁 ∧ (((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))) ↔ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ (((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))))
362		3anass 1042	. . . . . . . . . . . . . 14 ⊢ ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ (((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁)))
363	362	anbi2i 730	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁)) ↔ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ (((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))))
364	361, 363	bitr4i 267	. . . . . . . . . . . 12 ⊢ ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((2nd ‘𝑡) = 𝑁 ∧ (((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))) ↔ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁)))
365	360, 364	syl6bb 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (((2nd ‘𝑡) = 𝑁 ∧ (((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁)) ↔ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))))
366	365	notbid 308	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶) → (¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁)) ↔ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))))
367	366	pm5.32da 673	. . . . . . . . 9 ⊢ (𝜑 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁)))))
368	367	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁)))))
369	232, 349, 368	3bitr3d 298	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁)))))
370	369	rabbidva 3188	. . . . . 6 ⊢ (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁)))})
371		iunrab 4567	. . . . . 6 ⊢ ∪ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∃𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))(𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}
372		difrab 3901	. . . . . 6 ⊢ ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ¬ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁)))}
373	370, 371, 372	3eqtr4g 2681	. . . . 5 ⊢ (𝜑 → ∪ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} = ({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}))
374	373	fveq2d 6195	. . . 4 ⊢ (𝜑 → (#‘∪ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) = (#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))})))
375	27, 28	mp1i 13	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))) → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ∈ Fin)
376		simpl 473	. . . . . . . . . . . 12 ⊢ ((𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) → 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))))
377	376	a1i 11	. . . . . . . . . . 11 ⊢ (𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → ((𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾))) → 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
378	377	ss2rabi 3684	. . . . . . . . . 10 ⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
379	378	sseli 3599	. . . . . . . . 9 ⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} → 𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})
380		fveq2 6191	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑠 → (2nd ‘𝑡) = (2nd ‘𝑠))
381	380	breq2d 4665	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑠 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑠)))
382	381	ifbid 4108	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑠 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑠), 𝑦, (𝑦 + 1)))
383	382	csbeq1d 3540	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑠 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd ‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
384		fveq2 6191	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑠 → (1st ‘𝑡) = (1st ‘𝑠))
385	384	fveq2d 6195	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑠 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑠)))
386	384	fveq2d 6195	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑠 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑠)))
387	386	imaeq1d 5465	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑠 → ((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑠)) “ (1...𝑗)))
388	387	xpeq1d 5138	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑠 → (((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}))
389	386	imaeq1d 5465	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑠 → ((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)))
390	389	xpeq1d 5138	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑠 → (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0}))
391	388, 390	uneq12d 3768	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑠 → ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))
392	385, 391	oveq12d 6668	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑠 → ((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st ‘𝑠)) ∘𝑓 + ((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0}))))
393	392	csbeq2dv 3992	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑠 → ⦋if(𝑦 < (2nd ‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd ‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑠)) ∘𝑓 + ((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0}))))
394	383, 393	eqtrd 2656	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑠 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd ‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑠)) ∘𝑓 + ((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0}))))
395	394	mpteq2dv 4745	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑠 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑠)) ∘𝑓 + ((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))))
396	395	eqeq2d 2632	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑠 → (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑠)) ∘𝑓 + ((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
397		eqcom 2629	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑠)) ∘𝑓 + ((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑠)) ∘𝑓 + ((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥)
398	396, 397	syl6bb 276	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑠 → (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑠)) ∘𝑓 + ((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥))
399	398	elrab 3363	. . . . . . . . . 10 ⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ↔ (𝑠 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑠)) ∘𝑓 + ((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥))
400	399	simprbi 480	. . . . . . . . 9 ⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑠)) ∘𝑓 + ((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥)
401	379, 400	syl 17	. . . . . . . 8 ⊢ (𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑠)) ∘𝑓 + ((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥)
402	401	rgen 2922	. . . . . . 7 ⊢ ∀𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑠)) ∘𝑓 + ((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥
403	402	rgenw 2924	. . . . . 6 ⊢ ∀𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))∀𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑠)) ∘𝑓 + ((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥
404		invdisj 4638	. . . . . 6 ⊢ (∀𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))∀𝑠 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))} (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑠), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑠)) ∘𝑓 + ((((2nd ‘(1st ‘𝑠)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑠)) “ ((𝑗 + 1)...𝑁)) × {0})))) = 𝑥 → Disj 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})
405	403, 404	mp1i 13	. . . . 5 ⊢ (𝜑 → Disj 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))})
406	8, 375, 405	hashiun 14554	. . . 4 ⊢ (𝜑 → (#‘∪ 𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1))){𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) = Σ𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))(#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}))
407	374, 406	eqtr3d 2658	. . 3 ⊢ (𝜑 → (#‘({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶} ∖ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))})) = Σ𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))(#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}))
408		fo1st 7188	. . . . . . . . . . . . 13 ⊢ 1st :V–onto→V
409		fofun 6116	. . . . . . . . . . . . 13 ⊢ (1st :V–onto→V → Fun 1st )
410	408, 409	ax-mp 5	. . . . . . . . . . . 12 ⊢ Fun 1st
411		ssv 3625	. . . . . . . . . . . . 13 ⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ V
412		fof 6115	. . . . . . . . . . . . . . 15 ⊢ (1st :V–onto→V → 1st :V⟶V)
413	408, 412	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 1st :V⟶V
414	413	fdmi 6052	. . . . . . . . . . . . 13 ⊢ dom 1st = V
415	411, 414	sseqtr4i 3638	. . . . . . . . . . . 12 ⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ dom 1st
416		fores 6124	. . . . . . . . . . . 12 ⊢ ((Fun 1st ∧ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ dom 1st ) → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}))
417	410, 415, 416	mp2an 708	. . . . . . . . . . 11 ⊢ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))})
418		fveq2 6191	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑥 → (2nd ‘𝑡) = (2nd ‘𝑥))
419	418	eqeq1d 2624	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑥 → ((2nd ‘𝑡) = 𝑁 ↔ (2nd ‘𝑥) = 𝑁))
420		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑥 → (1st ‘𝑡) = (1st ‘𝑥))
421	420	csbeq1d 3540	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑥 → ⦋(1st ‘𝑡) / 𝑠⦌𝐶 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
422	421	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑥 → (𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
423	422	rexbidv 3052	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑥 → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
424	423	ralbidv 2986	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑥 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶))
425	420	fveq2d 6195	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑥 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑥)))
426	425	fveq1d 6193	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑥 → ((1st ‘(1st ‘𝑡))‘𝑁) = ((1st ‘(1st ‘𝑥))‘𝑁))
427	426	eqeq1d 2624	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑥 → (((1st ‘(1st ‘𝑡))‘𝑁) = 0 ↔ ((1st ‘(1st ‘𝑥))‘𝑁) = 0))
428	420	fveq2d 6195	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑥 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑥)))
429	428	fveq1d 6193	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑥 → ((2nd ‘(1st ‘𝑡))‘𝑁) = ((2nd ‘(1st ‘𝑥))‘𝑁))
430	429	eqeq1d 2624	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑥 → (((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁 ↔ ((2nd ‘(1st ‘𝑥))‘𝑁) = 𝑁))
431	424, 427, 430	3anbi123d 1399	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑥 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑥))‘𝑁) = 𝑁)))
432	419, 431	anbi12d 747	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑥 → (((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁)) ↔ ((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑥))‘𝑁) = 𝑁))))
433	432	rexrab 3370	. . . . . . . . . . . . . . 15 ⊢ (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠 ↔ ∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠))
434		xp1st 7198	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑥) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
435	434	anim1i 592	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑥))‘𝑁) = 𝑁)) → ((1st ‘𝑥) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑥))‘𝑁) = 𝑁)))
436		eleq1 2689	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1st ‘𝑥) = 𝑠 → ((1st ‘𝑥) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ↔ 𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})))
437		csbeq1a 3542	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑠 = (1st ‘𝑥) → 𝐶 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
438	437	eqcoms 2630	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((1st ‘𝑥) = 𝑠 → 𝐶 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶)
439	438	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((1st ‘𝑥) = 𝑠 → ⦋(1st ‘𝑥) / 𝑠⦌𝐶 = 𝐶)
440	439	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1st ‘𝑥) = 𝑠 → (𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑖 = 𝐶))
441	440	rexbidv 3052	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1st ‘𝑥) = 𝑠 → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶))
442	441	ralbidv 2986	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1st ‘𝑥) = 𝑠 → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶))
443		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1st ‘𝑥) = 𝑠 → (1st ‘(1st ‘𝑥)) = (1st ‘𝑠))
444	443	fveq1d 6193	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1st ‘𝑥) = 𝑠 → ((1st ‘(1st ‘𝑥))‘𝑁) = ((1st ‘𝑠)‘𝑁))
445	444	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1st ‘𝑥) = 𝑠 → (((1st ‘(1st ‘𝑥))‘𝑁) = 0 ↔ ((1st ‘𝑠)‘𝑁) = 0))
446		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1st ‘𝑥) = 𝑠 → (2nd ‘(1st ‘𝑥)) = (2nd ‘𝑠))
447	446	fveq1d 6193	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1st ‘𝑥) = 𝑠 → ((2nd ‘(1st ‘𝑥))‘𝑁) = ((2nd ‘𝑠)‘𝑁))
448	447	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1st ‘𝑥) = 𝑠 → (((2nd ‘(1st ‘𝑥))‘𝑁) = 𝑁 ↔ ((2nd ‘𝑠)‘𝑁) = 𝑁))
449	442, 445, 448	3anbi123d 1399	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1st ‘𝑥) = 𝑠 → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑥))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)))
450	436, 449	anbi12d 747	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘𝑥) = 𝑠 → (((1st ‘𝑥) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑥))‘𝑁) = 𝑁)) ↔ (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁))))
451	435, 450	syl5ibcom 235	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑥))‘𝑁) = 𝑁)) → ((1st ‘𝑥) = 𝑠 → (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁))))
452	451	adantrl 752	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ ((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑥))‘𝑁) = 𝑁))) → ((1st ‘𝑥) = 𝑠 → (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁))))
453	452	expimpd 629	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → ((((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠) → (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁))))
454	453	rexlimiv 3027	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠) → (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)))
455		nn0fz0 12437	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ0 ↔ 𝑁 ∈ (0...𝑁))
456	174, 455	sylib 208	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
457		opelxpi 5148	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑁 ∈ (0...𝑁)) → ⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
458	456, 457	sylan2 491	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝜑) → ⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
459	458	ancoms 469	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) → ⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
460		opelxp2 5151	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → 𝑁 ∈ (0...𝑁))
461		op2ndg 7181	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (2nd ‘⟨𝑠, 𝑁⟩) = 𝑁)
462	461	biantrurd 529	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁) ↔ ((2nd ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁))))
463		op1stg 7180	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (1st ‘⟨𝑠, 𝑁⟩) = 𝑠)
464		csbeq1a 3542	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑠 = (1st ‘⟨𝑠, 𝑁⟩) → 𝐶 = ⦋(1st ‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶)
465	464	eqcoms 2630	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((1st ‘⟨𝑠, 𝑁⟩) = 𝑠 → 𝐶 = ⦋(1st ‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶)
466	465	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((1st ‘⟨𝑠, 𝑁⟩) = 𝑠 → ⦋(1st ‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 = 𝐶)
467	463, 466	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ⦋(1st ‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 = 𝐶)
468	467	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (𝑖 = ⦋(1st ‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ↔ 𝑖 = 𝐶))
469	468	rexbidv 3052	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶))
470	469	ralbidv 2986	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶))
471	463	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (1st ‘(1st ‘⟨𝑠, 𝑁⟩)) = (1st ‘𝑠))
472	471	fveq1d 6193	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = ((1st ‘𝑠)‘𝑁))
473	472	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ↔ ((1st ‘𝑠)‘𝑁) = 0))
474	463	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (2nd ‘(1st ‘⟨𝑠, 𝑁⟩)) = (2nd ‘𝑠))
475	474	fveq1d 6193	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = ((2nd ‘𝑠)‘𝑁))
476	475	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁 ↔ ((2nd ‘𝑠)‘𝑁) = 𝑁))
477	470, 473, 476	3anbi123d 1399	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)))
478	463	biantrud 528	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → (((2nd ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ↔ (((2nd ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ∧ (1st ‘⟨𝑠, 𝑁⟩) = 𝑠)))
479	462, 477, 478	3bitr3d 298	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑠 ∈ V ∧ 𝑁 ∈ (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁) ↔ (((2nd ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ∧ (1st ‘⟨𝑠, 𝑁⟩) = 𝑠)))
480	44, 460, 479	sylancr 695	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁) ↔ (((2nd ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ∧ (1st ‘⟨𝑠, 𝑁⟩) = 𝑠)))
481	480	biimpa 501	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)) → (((2nd ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ∧ (1st ‘⟨𝑠, 𝑁⟩) = 𝑠))
482		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝑠, 𝑁⟩))
483	482	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → ((2nd ‘𝑥) = 𝑁 ↔ (2nd ‘⟨𝑠, 𝑁⟩) = 𝑁))
484		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → (1st ‘𝑥) = (1st ‘⟨𝑠, 𝑁⟩))
485	484	csbeq1d 3540	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → ⦋(1st ‘𝑥) / 𝑠⦌𝐶 = ⦋(1st ‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶)
486	485	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → (𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ 𝑖 = ⦋(1st ‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶))
487	486	rexbidv 3052	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → (∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ ∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶))
488	487	ralbidv 2986	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ↔ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶))
489	484	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘⟨𝑠, 𝑁⟩)))
490	489	fveq1d 6193	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → ((1st ‘(1st ‘𝑥))‘𝑁) = ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁))
491	490	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → (((1st ‘(1st ‘𝑥))‘𝑁) = 0 ↔ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0))
492	484	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘⟨𝑠, 𝑁⟩)))
493	492	fveq1d 6193	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → ((2nd ‘(1st ‘𝑥))‘𝑁) = ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁))
494	493	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → (((2nd ‘(1st ‘𝑥))‘𝑁) = 𝑁 ↔ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁))
495	488, 491, 494	3anbi123d 1399	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → ((∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑥))‘𝑁) = 𝑁) ↔ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)))
496	483, 495	anbi12d 747	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → (((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑥))‘𝑁) = 𝑁)) ↔ ((2nd ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁))))
497	484	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → ((1st ‘𝑥) = 𝑠 ↔ (1st ‘⟨𝑠, 𝑁⟩) = 𝑠))
498	496, 497	anbi12d 747	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = ⟨𝑠, 𝑁⟩ → ((((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠) ↔ (((2nd ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ∧ (1st ‘⟨𝑠, 𝑁⟩) = 𝑠)))
499	498	rspcev 3309	. . . . . . . . . . . . . . . . . . 19 ⊢ ((⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (((2nd ‘⟨𝑠, 𝑁⟩) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘⟨𝑠, 𝑁⟩) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘⟨𝑠, 𝑁⟩))‘𝑁) = 𝑁)) ∧ (1st ‘⟨𝑠, 𝑁⟩) = 𝑠)) → ∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠))
500	481, 499	syldan 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨𝑠, 𝑁⟩ ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)) → ∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠))
501	459, 500	sylan 488	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)) → ∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠))
502	501	expl 648	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)) → ∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠)))
503	454, 502	impbid2 216	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (∃𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))(((2nd ‘𝑥) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑥) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑥))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑥))‘𝑁) = 𝑁)) ∧ (1st ‘𝑥) = 𝑠) ↔ (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁))))
504	433, 503	syl5bb 272	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠 ↔ (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁))))
505	504	abbidv 2741	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠} = {𝑠 ∣ (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁))})
506		dfimafn 6245	. . . . . . . . . . . . . . 15 ⊢ ((Fun 1st ∧ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ dom 1st ) → (1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑦})
507	410, 415, 506	mp2an 708	. . . . . . . . . . . . . 14 ⊢ (1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑦}
508		nfv 1843	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑠(2nd ‘𝑡) = 𝑁
509		nfcv 2764	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑠(0...(𝑁 − 1))
510		nfcsb1v 3549	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑠⦋(1st ‘𝑡) / 𝑠⦌𝐶
511	510	nfeq2 2780	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑠 𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶
512	509, 511	nfrex 3007	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑠∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶
513	509, 512	nfral 2945	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑠∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶
514		nfv 1843	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑠((1st ‘(1st ‘𝑡))‘𝑁) = 0
515		nfv 1843	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑠((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁
516	513, 514, 515	nf3an 1831	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑠(∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁)
517	508, 516	nfan 1828	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑠((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))
518		nfcv 2764	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑠((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))
519	517, 518	nfrab 3123	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑠{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}
520		nfv 1843	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑠(1st ‘𝑥) = 𝑦
521	519, 520	nfrex 3007	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑠∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑦
522		nfv 1843	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠
523		eqeq2 2633	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑠 → ((1st ‘𝑥) = 𝑦 ↔ (1st ‘𝑥) = 𝑠))
524	523	rexbidv 3052	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑠 → (∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠))
525	521, 522, 524	cbvab 2746	. . . . . . . . . . . . . 14 ⊢ {𝑦 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑦} = {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠}
526	507, 525	eqtri 2644	. . . . . . . . . . . . 13 ⊢ (1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = {𝑠 ∣ ∃𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} (1st ‘𝑥) = 𝑠}
527		df-rab 2921	. . . . . . . . . . . . 13 ⊢ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} = {𝑠 ∣ (𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁))}
528	505, 526, 527	3eqtr4g 2681	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)})
529		foeq3 6113	. . . . . . . . . . . 12 ⊢ ((1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}) ↔ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}))
530	528, 529	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→(1st “ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}) ↔ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}))
531	417, 530	mpbii 223	. . . . . . . . . 10 ⊢ (𝜑 → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)})
532		fof 6115	. . . . . . . . . 10 ⊢ ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}⟶{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)})
533	531, 532	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}⟶{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)})
534		fvres 6207	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑥) = (1st ‘𝑥))
535		fvres 6207	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑦) = (1st ‘𝑦))
536	534, 535	eqeqan12d 2638	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}) → (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑦) ↔ (1st ‘𝑥) = (1st ‘𝑦)))
537		simpl 473	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁)) → (2nd ‘𝑡) = 𝑁)
538	537	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁)) → (2nd ‘𝑡) = 𝑁))
539	538	ss2rabi 3684	. . . . . . . . . . . . . 14 ⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} ⊆ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (2nd ‘𝑡) = 𝑁}
540	539	sseli 3599	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} → 𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (2nd ‘𝑡) = 𝑁})
541	419	elrab 3363	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (2nd ‘𝑡) = 𝑁} ↔ (𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (2nd ‘𝑥) = 𝑁))
542	540, 541	sylib 208	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} → (𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (2nd ‘𝑥) = 𝑁))
543	539	sseli 3599	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} → 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (2nd ‘𝑡) = 𝑁})
544		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑦 → (2nd ‘𝑡) = (2nd ‘𝑦))
545	544	eqeq1d 2624	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑦 → ((2nd ‘𝑡) = 𝑁 ↔ (2nd ‘𝑦) = 𝑁))
546	545	elrab 3363	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (2nd ‘𝑡) = 𝑁} ↔ (𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (2nd ‘𝑦) = 𝑁))
547	543, 546	sylib 208	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} → (𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (2nd ‘𝑦) = 𝑁))
548		eqtr3 2643	. . . . . . . . . . . . . 14 ⊢ (((2nd ‘𝑥) = 𝑁 ∧ (2nd ‘𝑦) = 𝑁) → (2nd ‘𝑥) = (2nd ‘𝑦))
549		xpopth 7207	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) = (2nd ‘𝑦)) ↔ 𝑥 = 𝑦))
550	549	biimpd 219	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) = (2nd ‘𝑦)) → 𝑥 = 𝑦))
551	550	ancomsd 470	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (((2nd ‘𝑥) = (2nd ‘𝑦) ∧ (1st ‘𝑥) = (1st ‘𝑦)) → 𝑥 = 𝑦))
552	551	expdimp 453	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) ∧ (2nd ‘𝑥) = (2nd ‘𝑦)) → ((1st ‘𝑥) = (1st ‘𝑦) → 𝑥 = 𝑦))
553	548, 552	sylan2 491	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) ∧ ((2nd ‘𝑥) = 𝑁 ∧ (2nd ‘𝑦) = 𝑁)) → ((1st ‘𝑥) = (1st ‘𝑦) → 𝑥 = 𝑦))
554	553	an4s 869	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (2nd ‘𝑥) = 𝑁) ∧ (𝑦 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ (2nd ‘𝑦) = 𝑁)) → ((1st ‘𝑥) = (1st ‘𝑦) → 𝑥 = 𝑦))
555	542, 547, 554	syl2an 494	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}) → ((1st ‘𝑥) = (1st ‘𝑦) → 𝑥 = 𝑦))
556	536, 555	sylbid 230	. . . . . . . . . 10 ⊢ ((𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} ∧ 𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}) → (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑦) → 𝑥 = 𝑦))
557	556	rgen2a 2977	. . . . . . . . 9 ⊢ ∀𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}∀𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑦) → 𝑥 = 𝑦)
558	533, 557	jctir 561	. . . . . . . 8 ⊢ (𝜑 → ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}⟶{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} ∧ ∀𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}∀𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑦) → 𝑥 = 𝑦)))
559		dff13 6512	. . . . . . . 8 ⊢ ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}–1-1→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} ↔ ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}⟶{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} ∧ ∀𝑥 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}∀𝑦 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} (((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑥) = ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))})‘𝑦) → 𝑥 = 𝑦)))
560	558, 559	sylibr 224	. . . . . . 7 ⊢ (𝜑 → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}–1-1→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)})
561		df-f1o 5895	. . . . . . 7 ⊢ ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} ↔ ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}–1-1→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} ∧ (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}–onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}))
562	560, 531, 561	sylanbrc 698	. . . . . 6 ⊢ (𝜑 → (1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)})
563		rabfi 8185	. . . . . . . . 9 ⊢ (((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∈ Fin → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} ∈ Fin)
564	27, 563	ax-mp 5	. . . . . . . 8 ⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} ∈ Fin
565	564	elexi 3213	. . . . . . 7 ⊢ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} ∈ V
566	565	f1oen 7976	. . . . . 6 ⊢ ((1st ↾ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}):{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)})
567	562, 566	syl 17	. . . . 5 ⊢ (𝜑 → {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)})
568		rabfi 8185	. . . . . . 7 ⊢ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∈ Fin → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} ∈ Fin)
569	24, 568	ax-mp 5	. . . . . 6 ⊢ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} ∈ Fin
570		hashen 13135	. . . . . 6 ⊢ (({𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} ∈ Fin ∧ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)} ∈ Fin) → ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}) ↔ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}))
571	564, 569, 570	mp2an 708	. . . . 5 ⊢ ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}) ↔ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)})
572	567, 571	sylibr 224	. . . 4 ⊢ (𝜑 → (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))}) = (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)}))
573	572	oveq2d 6666	. . 3 ⊢ (𝜑 → ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) − (#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ((2nd ‘𝑡) = 𝑁 ∧ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶 ∧ ((1st ‘(1st ‘𝑡))‘𝑁) = 0 ∧ ((2nd ‘(1st ‘𝑡))‘𝑁) = 𝑁))})) = ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)})))
574	204, 407, 573	3eqtr3d 2664	. 2 ⊢ (𝜑 → Σ𝑥 ∈ (((0...𝐾) ↑𝑚 (1...𝑁)) ↑𝑚 (0...(𝑁 − 1)))(#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ (𝑥 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘𝑓 + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ∧ ((0...(𝑁 − 1)) ⊆ ran (𝑝 ∈ ran 𝑥 ↦ 𝐵) ∧ ∀𝑛 ∈ (1...𝑁)(∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 0 ∧ ∃𝑝 ∈ ran 𝑥(𝑝‘𝑛) ≠ 𝐾)))}) = ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)})))
575	167, 574	breqtrd 4679	1 ⊢ (𝜑 → 2 ∥ ((#‘{𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ ∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ ((0...𝑁) ∖ {(2nd ‘𝑡)})𝑖 = ⦋(1st ‘𝑡) / 𝑠⦌𝐶}) − (#‘{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∣ (∀𝑖 ∈ (0...(𝑁 − 1))∃𝑗 ∈ (0...(𝑁 − 1))𝑖 = 𝐶 ∧ ((1st ‘𝑠)‘𝑁) = 0 ∧ ((2nd ‘𝑠)‘𝑁) = 𝑁)})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃!weu 2470  {cab 2608   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  ∃!wreu 2914  {crab 2916  Vcvv 3200  [wsbc 3435  ⦋csb 3533   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  ifcif 4086  {csn 4177  ⟨cop 4183  ∪ ciun 4520  Disj wdisj 4620   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112  dom cdm 5114  ran crn 5115   ↾ cres 5116   “ cima 5117  Fun wfun 5882  ⟶wf 5884  –1-1→wf1 5885  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895  1^st c1st 7166  2^nd c2nd 7167   ↑_𝑚 cmap 7857   ≈ cen 7952  Fincfn 7955  ℂcc 9934  0cc0 9936  1c1 9937   + caddc 9939   < clt 10074   − cmin 10266  ℕcn 11020  2c2 11070  ℕ₀cn0 11292  ℤcz 11377  ℤ_≥cuz 11687  ...cfz 12326  ..^cfzo 12465  #chash 13117  Σcsu 14416   ∥ cdvds 14983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984

This theorem is referenced by:  poimirlem28  33437