Theorem vdwlem13 15697

Description: Lemma for vdw 15698. Main induction on 𝐾; 𝐾 = 0, 𝐾 = 1 base cases. (Contributed by Mario Carneiro, 18-Aug-2014.)

Hypotheses

Ref	Expression
vdw.r	⊢ (𝜑 → 𝑅 ∈ Fin)
vdw.k	⊢ (𝜑 → 𝐾 ∈ ℕ0)

Assertion

Ref	Expression
vdwlem13	⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓)

Distinct variable groups:   𝜑,𝑛,𝑓   𝑓,𝐾,𝑛   𝑅,𝑓,𝑛   𝜑,𝑓

Proof of Theorem vdwlem13

Dummy variables 𝑎 𝑐 𝑑 𝑔 𝑘 𝑚 𝑥 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnn1uz2 11765	. . 3 ⊢ (𝐾 ∈ ℕ ↔ (𝐾 = 1 ∨ 𝐾 ∈ (ℤ≥‘2)))
2		vdw.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Fin)
3		ovex 6678	. . . . . . . . . 10 ⊢ (1...1) ∈ V
4		elmapg 7870	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Fin ∧ (1...1) ∈ V) → (𝑓 ∈ (𝑅 ↑𝑚 (1...1)) ↔ 𝑓:(1...1)⟶𝑅))
5	2, 3, 4	sylancl 694	. . . . . . . . 9 ⊢ (𝜑 → (𝑓 ∈ (𝑅 ↑𝑚 (1...1)) ↔ 𝑓:(1...1)⟶𝑅))
6	5	biimpa 501	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑅 ↑𝑚 (1...1))) → 𝑓:(1...1)⟶𝑅)
7		1nn 11031	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
8		vdwap1 15681	. . . . . . . . . 10 ⊢ ((1 ∈ ℕ ∧ 1 ∈ ℕ) → (1(AP‘1)1) = {1})
9	7, 7, 8	mp2an 708	. . . . . . . . 9 ⊢ (1(AP‘1)1) = {1}
10		1z 11407	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
11		elfz3 12351	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → 1 ∈ (1...1))
12	10, 11	mp1i 13	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → 1 ∈ (1...1))
13		eqidd 2623	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → (𝑓‘1) = (𝑓‘1))
14		ffn 6045	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...1)⟶𝑅 → 𝑓 Fn (1...1))
15	14	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → 𝑓 Fn (1...1))
16		fniniseg 6338	. . . . . . . . . . . 12 ⊢ (𝑓 Fn (1...1) → (1 ∈ (◡𝑓 “ {(𝑓‘1)}) ↔ (1 ∈ (1...1) ∧ (𝑓‘1) = (𝑓‘1))))
17	15, 16	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → (1 ∈ (◡𝑓 “ {(𝑓‘1)}) ↔ (1 ∈ (1...1) ∧ (𝑓‘1) = (𝑓‘1))))
18	12, 13, 17	mpbir2and 957	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → 1 ∈ (◡𝑓 “ {(𝑓‘1)}))
19	18	snssd 4340	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → {1} ⊆ (◡𝑓 “ {(𝑓‘1)}))
20	9, 19	syl5eqss 3649	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓:(1...1)⟶𝑅) → (1(AP‘1)1) ⊆ (◡𝑓 “ {(𝑓‘1)}))
21	6, 20	syldan 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑅 ↑𝑚 (1...1))) → (1(AP‘1)1) ⊆ (◡𝑓 “ {(𝑓‘1)}))
22	21	ralrimiva 2966	. . . . . 6 ⊢ (𝜑 → ∀𝑓 ∈ (𝑅 ↑𝑚 (1...1))(1(AP‘1)1) ⊆ (◡𝑓 “ {(𝑓‘1)}))
23		fveq2 6191	. . . . . . . . 9 ⊢ (𝐾 = 1 → (AP‘𝐾) = (AP‘1))
24	23	oveqd 6667	. . . . . . . 8 ⊢ (𝐾 = 1 → (1(AP‘𝐾)1) = (1(AP‘1)1))
25	24	sseq1d 3632	. . . . . . 7 ⊢ (𝐾 = 1 → ((1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘1)1) ⊆ (◡𝑓 “ {(𝑓‘1)})))
26	25	ralbidv 2986	. . . . . 6 ⊢ (𝐾 = 1 → (∀𝑓 ∈ (𝑅 ↑𝑚 (1...1))(1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}) ↔ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...1))(1(AP‘1)1) ⊆ (◡𝑓 “ {(𝑓‘1)})))
27	22, 26	syl5ibrcom 237	. . . . 5 ⊢ (𝜑 → (𝐾 = 1 → ∀𝑓 ∈ (𝑅 ↑𝑚 (1...1))(1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)})))
28		oveq1 6657	. . . . . . . . . . . 12 ⊢ (𝑎 = 1 → (𝑎(AP‘𝐾)𝑑) = (1(AP‘𝐾)𝑑))
29	28	sseq1d 3632	. . . . . . . . . . 11 ⊢ (𝑎 = 1 → ((𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)})))
30		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑑 = 1 → (1(AP‘𝐾)𝑑) = (1(AP‘𝐾)1))
31	30	sseq1d 3632	. . . . . . . . . . 11 ⊢ (𝑑 = 1 → ((1(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)}) ↔ (1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)})))
32	29, 31	rspc2ev 3324	. . . . . . . . . 10 ⊢ ((1 ∈ ℕ ∧ 1 ∈ ℕ ∧ (1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)})) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)}))
33	7, 7, 32	mp3an12 1414	. . . . . . . . 9 ⊢ ((1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)}))
34		fvex 6201	. . . . . . . . . 10 ⊢ (𝑓‘1) ∈ V
35		sneq 4187	. . . . . . . . . . . . 13 ⊢ (𝑐 = (𝑓‘1) → {𝑐} = {(𝑓‘1)})
36	35	imaeq2d 5466	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝑓‘1) → (◡𝑓 “ {𝑐}) = (◡𝑓 “ {(𝑓‘1)}))
37	36	sseq2d 3633	. . . . . . . . . . 11 ⊢ (𝑐 = (𝑓‘1) → ((𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {𝑐}) ↔ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)})))
38	37	2rexbidv 3057	. . . . . . . . . 10 ⊢ (𝑐 = (𝑓‘1) → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {𝑐}) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)})))
39	34, 38	spcev 3300	. . . . . . . . 9 ⊢ (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {(𝑓‘1)}) → ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {𝑐}))
40	33, 39	syl 17	. . . . . . . 8 ⊢ ((1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}) → ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {𝑐}))
41		vdw.k	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
42	41	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑅 ↑𝑚 (1...1))) → 𝐾 ∈ ℕ0)
43	3, 42, 6	vdwmc 15682	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑅 ↑𝑚 (1...1))) → (𝐾 MonoAP 𝑓 ↔ ∃𝑐∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ (𝑎(AP‘𝐾)𝑑) ⊆ (◡𝑓 “ {𝑐})))
44	40, 43	syl5ibr 236	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑅 ↑𝑚 (1...1))) → ((1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}) → 𝐾 MonoAP 𝑓))
45	44	ralimdva 2962	. . . . . 6 ⊢ (𝜑 → (∀𝑓 ∈ (𝑅 ↑𝑚 (1...1))(1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}) → ∀𝑓 ∈ (𝑅 ↑𝑚 (1...1))𝐾 MonoAP 𝑓))
46		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑛 = 1 → (1...𝑛) = (1...1))
47	46	oveq2d 6666	. . . . . . . . 9 ⊢ (𝑛 = 1 → (𝑅 ↑𝑚 (1...𝑛)) = (𝑅 ↑𝑚 (1...1)))
48	47	raleqdv 3144	. . . . . . . 8 ⊢ (𝑛 = 1 → (∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...1))𝐾 MonoAP 𝑓))
49	48	rspcev 3309	. . . . . . 7 ⊢ ((1 ∈ ℕ ∧ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...1))𝐾 MonoAP 𝑓) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓)
50	7, 49	mpan 706	. . . . . 6 ⊢ (∀𝑓 ∈ (𝑅 ↑𝑚 (1...1))𝐾 MonoAP 𝑓 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓)
51	45, 50	syl6 35	. . . . 5 ⊢ (𝜑 → (∀𝑓 ∈ (𝑅 ↑𝑚 (1...1))(1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
52	27, 51	syld 47	. . . 4 ⊢ (𝜑 → (𝐾 = 1 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
53		breq1 4656	. . . . . . . 8 ⊢ (𝑥 = 2 → (𝑥 MonoAP 𝑓 ↔ 2 MonoAP 𝑓))
54	53	rexralbidv 3058	. . . . . . 7 ⊢ (𝑥 = 2 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))2 MonoAP 𝑓))
55	54	ralbidv 2986	. . . . . 6 ⊢ (𝑥 = 2 → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))2 MonoAP 𝑓))
56		breq1 4656	. . . . . . . 8 ⊢ (𝑥 = 𝑘 → (𝑥 MonoAP 𝑓 ↔ 𝑘 MonoAP 𝑓))
57	56	rexralbidv 3058	. . . . . . 7 ⊢ (𝑥 = 𝑘 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓))
58	57	ralbidv 2986	. . . . . 6 ⊢ (𝑥 = 𝑘 → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓))
59		breq1 4656	. . . . . . . 8 ⊢ (𝑥 = (𝑘 + 1) → (𝑥 MonoAP 𝑓 ↔ (𝑘 + 1) MonoAP 𝑓))
60	59	rexralbidv 3058	. . . . . . 7 ⊢ (𝑥 = (𝑘 + 1) → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))(𝑘 + 1) MonoAP 𝑓))
61	60	ralbidv 2986	. . . . . 6 ⊢ (𝑥 = (𝑘 + 1) → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))(𝑘 + 1) MonoAP 𝑓))
62		breq1 4656	. . . . . . . 8 ⊢ (𝑥 = 𝐾 → (𝑥 MonoAP 𝑓 ↔ 𝐾 MonoAP 𝑓))
63	62	rexralbidv 3058	. . . . . . 7 ⊢ (𝑥 = 𝐾 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
64	63	ralbidv 2986	. . . . . 6 ⊢ (𝑥 = 𝐾 → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑥 MonoAP 𝑓 ↔ ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
65		hashcl 13147	. . . . . . . . . 10 ⊢ (𝑟 ∈ Fin → (#‘𝑟) ∈ ℕ0)
66		nn0p1nn 11332	. . . . . . . . . 10 ⊢ ((#‘𝑟) ∈ ℕ0 → ((#‘𝑟) + 1) ∈ ℕ)
67	65, 66	syl 17	. . . . . . . . 9 ⊢ (𝑟 ∈ Fin → ((#‘𝑟) + 1) ∈ ℕ)
68		simpll 790	. . . . . . . . . . . 12 ⊢ (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑𝑚 (1...((#‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → 𝑟 ∈ Fin)
69		simplr 792	. . . . . . . . . . . . 13 ⊢ (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑𝑚 (1...((#‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → 𝑓 ∈ (𝑟 ↑𝑚 (1...((#‘𝑟) + 1))))
70		vex 3203	. . . . . . . . . . . . . 14 ⊢ 𝑟 ∈ V
71		ovex 6678	. . . . . . . . . . . . . 14 ⊢ (1...((#‘𝑟) + 1)) ∈ V
72	70, 71	elmap 7886	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝑟 ↑𝑚 (1...((#‘𝑟) + 1))) ↔ 𝑓:(1...((#‘𝑟) + 1))⟶𝑟)
73	69, 72	sylib 208	. . . . . . . . . . . 12 ⊢ (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑𝑚 (1...((#‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → 𝑓:(1...((#‘𝑟) + 1))⟶𝑟)
74		simpr 477	. . . . . . . . . . . 12 ⊢ (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑𝑚 (1...((#‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓) → ¬ 2 MonoAP 𝑓)
75	68, 73, 74	vdwlem12 15696	. . . . . . . . . . 11 ⊢ ¬ ((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑𝑚 (1...((#‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓)
76		iman 440	. . . . . . . . . . 11 ⊢ (((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑𝑚 (1...((#‘𝑟) + 1)))) → 2 MonoAP 𝑓) ↔ ¬ ((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑𝑚 (1...((#‘𝑟) + 1)))) ∧ ¬ 2 MonoAP 𝑓))
77	75, 76	mpbir 221	. . . . . . . . . 10 ⊢ ((𝑟 ∈ Fin ∧ 𝑓 ∈ (𝑟 ↑𝑚 (1...((#‘𝑟) + 1)))) → 2 MonoAP 𝑓)
78	77	ralrimiva 2966	. . . . . . . . 9 ⊢ (𝑟 ∈ Fin → ∀𝑓 ∈ (𝑟 ↑𝑚 (1...((#‘𝑟) + 1)))2 MonoAP 𝑓)
79		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑛 = ((#‘𝑟) + 1) → (1...𝑛) = (1...((#‘𝑟) + 1)))
80	79	oveq2d 6666	. . . . . . . . . . 11 ⊢ (𝑛 = ((#‘𝑟) + 1) → (𝑟 ↑𝑚 (1...𝑛)) = (𝑟 ↑𝑚 (1...((#‘𝑟) + 1))))
81	80	raleqdv 3144	. . . . . . . . . 10 ⊢ (𝑛 = ((#‘𝑟) + 1) → (∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))2 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...((#‘𝑟) + 1)))2 MonoAP 𝑓))
82	81	rspcev 3309	. . . . . . . . 9 ⊢ ((((#‘𝑟) + 1) ∈ ℕ ∧ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...((#‘𝑟) + 1)))2 MonoAP 𝑓) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))2 MonoAP 𝑓)
83	67, 78, 82	syl2anc 693	. . . . . . . 8 ⊢ (𝑟 ∈ Fin → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))2 MonoAP 𝑓)
84	83	rgen 2922	. . . . . . 7 ⊢ ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))2 MonoAP 𝑓
85	84	a1i 11	. . . . . 6 ⊢ (2 ∈ ℤ → ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))2 MonoAP 𝑓)
86		oveq1 6657	. . . . . . . . . . 11 ⊢ (𝑟 = 𝑠 → (𝑟 ↑𝑚 (1...𝑛)) = (𝑠 ↑𝑚 (1...𝑛)))
87	86	raleqdv 3144	. . . . . . . . . 10 ⊢ (𝑟 = 𝑠 → (∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑠 ↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓))
88	87	rexbidv 3052	. . . . . . . . 9 ⊢ (𝑟 = 𝑠 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓))
89		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚))
90	89	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝑠 ↑𝑚 (1...𝑛)) = (𝑠 ↑𝑚 (1...𝑚)))
91	90	raleqdv 3144	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (∀𝑓 ∈ (𝑠 ↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑓))
92		breq2 4657	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (𝑘 MonoAP 𝑓 ↔ 𝑘 MonoAP 𝑔))
93	92	cbvralv 3171	. . . . . . . . . . 11 ⊢ (∀𝑓 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑓 ↔ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔)
94	91, 93	syl6bb 276	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (∀𝑓 ∈ (𝑠 ↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔))
95	94	cbvrexv 3172	. . . . . . . . 9 ⊢ (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔)
96	88, 95	syl6bb 276	. . . . . . . 8 ⊢ (𝑟 = 𝑠 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔))
97	96	cbvralv 3171	. . . . . . 7 ⊢ (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔)
98		simplr 792	. . . . . . . . . 10 ⊢ (((𝑘 ∈ (ℤ≥‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔) → 𝑟 ∈ Fin)
99		simpll 790	. . . . . . . . . 10 ⊢ (((𝑘 ∈ (ℤ≥‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔) → 𝑘 ∈ (ℤ≥‘2))
100		simpr 477	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ (ℤ≥‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔) → ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔)
101	95	ralbii 2980	. . . . . . . . . . 11 ⊢ (∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓 ↔ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔)
102	100, 101	sylibr 224	. . . . . . . . . 10 ⊢ (((𝑘 ∈ (ℤ≥‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔) → ∀𝑠 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑠 ↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓)
103	98, 99, 102	vdwlem11 15695	. . . . . . . . 9 ⊢ (((𝑘 ∈ (ℤ≥‘2) ∧ 𝑟 ∈ Fin) ∧ ∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))(𝑘 + 1) MonoAP 𝑓)
104	103	ex 450	. . . . . . . 8 ⊢ ((𝑘 ∈ (ℤ≥‘2) ∧ 𝑟 ∈ Fin) → (∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))(𝑘 + 1) MonoAP 𝑓))
105	104	ralrimdva 2969	. . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘2) → (∀𝑠 ∈ Fin ∃𝑚 ∈ ℕ ∀𝑔 ∈ (𝑠 ↑𝑚 (1...𝑚))𝑘 MonoAP 𝑔 → ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))(𝑘 + 1) MonoAP 𝑓))
106	97, 105	syl5bi 232	. . . . . 6 ⊢ (𝑘 ∈ (ℤ≥‘2) → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝑘 MonoAP 𝑓 → ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))(𝑘 + 1) MonoAP 𝑓))
107	55, 58, 61, 64, 85, 106	uzind4 11746	. . . . 5 ⊢ (𝐾 ∈ (ℤ≥‘2) → ∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓)
108		oveq1 6657	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑟 ↑𝑚 (1...𝑛)) = (𝑅 ↑𝑚 (1...𝑛)))
109	108	raleqdv 3144	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
110	109	rexbidv 3052	. . . . . 6 ⊢ (𝑟 = 𝑅 → (∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓 ↔ ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
111	110	rspcv 3305	. . . . 5 ⊢ (𝑅 ∈ Fin → (∀𝑟 ∈ Fin ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑟 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
112	2, 107, 111	syl2im 40	. . . 4 ⊢ (𝜑 → (𝐾 ∈ (ℤ≥‘2) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
113	52, 112	jaod 395	. . 3 ⊢ (𝜑 → ((𝐾 = 1 ∨ 𝐾 ∈ (ℤ≥‘2)) → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
114	1, 113	syl5bi 232	. 2 ⊢ (𝜑 → (𝐾 ∈ ℕ → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
115		fveq2 6191	. . . . . . 7 ⊢ (𝐾 = 0 → (AP‘𝐾) = (AP‘0))
116	115	oveqd 6667	. . . . . 6 ⊢ (𝐾 = 0 → (1(AP‘𝐾)1) = (1(AP‘0)1))
117		vdwap0 15680	. . . . . . 7 ⊢ ((1 ∈ ℕ ∧ 1 ∈ ℕ) → (1(AP‘0)1) = ∅)
118	7, 7, 117	mp2an 708	. . . . . 6 ⊢ (1(AP‘0)1) = ∅
119	116, 118	syl6eq 2672	. . . . 5 ⊢ (𝐾 = 0 → (1(AP‘𝐾)1) = ∅)
120		0ss 3972	. . . . 5 ⊢ ∅ ⊆ (◡𝑓 “ {(𝑓‘1)})
121	119, 120	syl6eqss 3655	. . . 4 ⊢ (𝐾 = 0 → (1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}))
122	121	ralrimivw 2967	. . 3 ⊢ (𝐾 = 0 → ∀𝑓 ∈ (𝑅 ↑𝑚 (1...1))(1(AP‘𝐾)1) ⊆ (◡𝑓 “ {(𝑓‘1)}))
123	122, 51	syl5 34	. 2 ⊢ (𝜑 → (𝐾 = 0 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓))
124		elnn0 11294	. . 3 ⊢ (𝐾 ∈ ℕ0 ↔ (𝐾 ∈ ℕ ∨ 𝐾 = 0))
125	41, 124	sylib 208	. 2 ⊢ (𝜑 → (𝐾 ∈ ℕ ∨ 𝐾 = 0))
126	114, 123, 125	mpjaod 396	1 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑓 ∈ (𝑅 ↑𝑚 (1...𝑛))𝐾 MonoAP 𝑓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  {csn 4177   class class class wbr 4653  ^◡ccnv 5113   “ cima 5117   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  Fincfn 7955  0cc0 9936  1c1 9937   + caddc 9939  ℕcn 11020  2c2 11070  ℕ₀cn0 11292  ℤcz 11377  ℤ_≥cuz 11687  ...cfz 12326  #chash 13117  APcvdwa 15669   MonoAP cvdwm 15670

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-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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  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-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-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-hash 13118  df-vdwap 15672  df-vdwmc 15673  df-vdwpc 15674

This theorem is referenced by:  vdw  15698