Theorem fi1uzindOLD 13285

Description: Obsolete version of fi1uzind 13279 as of 28-Mar-2021. (Contributed by AV, 22-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
fi1uzindOLD.f	⊢ 𝐹 ∈ 𝑈
fi1uzindOLD.l	⊢ 𝐿 ∈ ℕ0
fi1uzindOLD.1	⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑))
fi1uzindOLD.2	⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃))
fi1uzindOLD.3	⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 ∈ 𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌)
fi1uzindOLD.4	⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒))
fi1uzindOLD.base	⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = 𝐿) → 𝜓)
fi1uzindOLD.step	⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)

Assertion

Ref	Expression
fi1uzindOLD	⊢ (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)

Distinct variable groups:   𝑎,𝑏,𝑒,𝑛,𝑣,𝑦,𝑓,𝑤   𝐸,𝑎,𝑒,𝑛,𝑣   𝐹,𝑎,𝑓,𝑤   𝑒,𝐿,𝑛,𝑣,𝑦   𝑉,𝑎,𝑏,𝑒,𝑛,𝑣   𝜓,𝑓,𝑛,𝑤,𝑦   𝜃,𝑒,𝑛,𝑣   𝜒,𝑓,𝑤   𝜑,𝑒,𝑛,𝑣   𝜌,𝑒,𝑓,𝑛,𝑣,𝑤,𝑦

Allowed substitution hints:   𝜑(𝑦,𝑤,𝑓,𝑎,𝑏)   𝜓(𝑣,𝑒,𝑎,𝑏)   𝜒(𝑦,𝑣,𝑒,𝑛,𝑎,𝑏)   𝜃(𝑦,𝑤,𝑓,𝑎,𝑏)   𝜌(𝑎,𝑏)   𝑈(𝑦,𝑤,𝑣,𝑒,𝑓,𝑛,𝑎,𝑏)   𝐸(𝑦,𝑤,𝑓,𝑏)   𝐹(𝑦,𝑣,𝑒,𝑛,𝑏)   𝐿(𝑤,𝑓,𝑎,𝑏)   𝑉(𝑦,𝑤,𝑓)

Proof of Theorem fi1uzindOLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hashcl 13147	. . . 4 ⊢ (𝑉 ∈ Fin → (#‘𝑉) ∈ ℕ0)
2		df-clel 2618	. . . . 5 ⊢ ((#‘𝑉) ∈ ℕ0 ↔ ∃𝑛(𝑛 = (#‘𝑉) ∧ 𝑛 ∈ ℕ0))
3		fi1uzindOLD.l	. . . . . . . . . . . . . . 15 ⊢ 𝐿 ∈ ℕ0
4		nn0z 11400	. . . . . . . . . . . . . . 15 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℤ)
5	3, 4	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (#‘𝑉)) → 𝐿 ∈ ℤ)
6		nn0z 11400	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℤ)
7	6	ad2antlr 763	. . . . . . . . . . . . . 14 ⊢ (((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (#‘𝑉)) → 𝑛 ∈ ℤ)
8		breq2 4657	. . . . . . . . . . . . . . . . . 18 ⊢ ((#‘𝑉) = 𝑛 → (𝐿 ≤ (#‘𝑉) ↔ 𝐿 ≤ 𝑛))
9	8	eqcoms 2630	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (#‘𝑉) → (𝐿 ≤ (#‘𝑉) ↔ 𝐿 ≤ 𝑛))
10	9	biimpcd 239	. . . . . . . . . . . . . . . 16 ⊢ (𝐿 ≤ (#‘𝑉) → (𝑛 = (#‘𝑉) → 𝐿 ≤ 𝑛))
11	10	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) → (𝑛 = (#‘𝑉) → 𝐿 ≤ 𝑛))
12	11	imp 445	. . . . . . . . . . . . . 14 ⊢ (((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (#‘𝑉)) → 𝐿 ≤ 𝑛)
13		eqeq1 2626	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝐿 → (𝑥 = (#‘𝑣) ↔ 𝐿 = (#‘𝑣)))
14	13	anbi2d 740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐿 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (#‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝐿 = (#‘𝑣))))
15	14	imbi1d 331	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐿 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (#‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝐿 = (#‘𝑣)) → 𝜓)))
16	15	2albidv 1851	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐿 → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (#‘𝑣)) → 𝜓) ↔ ∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝐿 = (#‘𝑣)) → 𝜓)))
17		eqeq1 2626	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝑥 = (#‘𝑣) ↔ 𝑦 = (#‘𝑣)))
18	17	anbi2d 740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (#‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑣))))
19	18	imbi1d 331	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (#‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑣)) → 𝜓)))
20	19	2albidv 1851	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (#‘𝑣)) → 𝜓) ↔ ∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑣)) → 𝜓)))
21		eqeq1 2626	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 = (#‘𝑣) ↔ (𝑦 + 1) = (#‘𝑣)))
22	21	anbi2d 740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 + 1) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (#‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣))))
23	22	imbi1d 331	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 + 1) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (#‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓)))
24	23	2albidv 1851	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑦 + 1) → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (#‘𝑣)) → 𝜓) ↔ ∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓)))
25		eqeq1 2626	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑛 → (𝑥 = (#‘𝑣) ↔ 𝑛 = (#‘𝑣)))
26	25	anbi2d 740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑛 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (#‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑣))))
27	26	imbi1d 331	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑛 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (#‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑣)) → 𝜓)))
28	27	2albidv 1851	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑛 → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (#‘𝑣)) → 𝜓) ↔ ∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑣)) → 𝜓)))
29		eqcom 2629	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐿 = (#‘𝑣) ↔ (#‘𝑣) = 𝐿)
30		fi1uzindOLD.base	. . . . . . . . . . . . . . . . . 18 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = 𝐿) → 𝜓)
31	29, 30	sylan2b 492	. . . . . . . . . . . . . . . . 17 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝐿 = (#‘𝑣)) → 𝜓)
32	31	gen2 1723	. . . . . . . . . . . . . . . 16 ⊢ ∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝐿 = (#‘𝑣)) → 𝜓)
33	32	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝐿 ∈ ℤ → ∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝐿 = (#‘𝑣)) → 𝜓))
34		simpl 473	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → 𝑣 = 𝑤)
35		simpr 477	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → 𝑒 = 𝑓)
36	35	sbceq1d 3440	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ([𝑒 / 𝑏]𝜌 ↔ [𝑓 / 𝑏]𝜌))
37	34, 36	sbceqbid 3442	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ↔ [𝑤 / 𝑎][𝑓 / 𝑏]𝜌))
38		fveq2 6191	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = 𝑤 → (#‘𝑣) = (#‘𝑤))
39	38	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝑤 → (𝑦 = (#‘𝑣) ↔ 𝑦 = (#‘𝑤)))
40	39	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝑦 = (#‘𝑣) ↔ 𝑦 = (#‘𝑤)))
41	37, 40	anbi12d 747	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑣)) ↔ ([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤))))
42		fi1uzindOLD.2	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃))
43	41, 42	imbi12d 334	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑣)) → 𝜓) ↔ (([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃)))
44	43	cbval2v 2285	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑣)) → 𝜓) ↔ ∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃))
45		nn0ge0 11318	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐿 ∈ ℕ0 → 0 ≤ 𝐿)
46		0red 10041	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ ℤ → 0 ∈ ℝ)
47		nn0re 11301	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℝ)
48	3, 47	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ ℤ → 𝐿 ∈ ℝ)
49		zre 11381	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℝ)
50		letr 10131	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((0 ∈ ℝ ∧ 𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((0 ≤ 𝐿 ∧ 𝐿 ≤ 𝑦) → 0 ≤ 𝑦))
51	46, 48, 49, 50	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ℤ → ((0 ≤ 𝐿 ∧ 𝐿 ≤ 𝑦) → 0 ≤ 𝑦))
52		0nn0 11307	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ 0 ∈ ℕ0
53		pm3.22 465	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((0 ≤ 𝑦 ∧ 𝑦 ∈ ℤ) → (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦))
54		0z 11388	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ 0 ∈ ℤ
55		eluz1 11691	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (0 ∈ ℤ → (𝑦 ∈ (ℤ≥‘0) ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦)))
56	54, 55	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((0 ≤ 𝑦 ∧ 𝑦 ∈ ℤ) → (𝑦 ∈ (ℤ≥‘0) ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦)))
57	53, 56	mpbird 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((0 ≤ 𝑦 ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ (ℤ≥‘0))
58		eluznn0 11757	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((0 ∈ ℕ0 ∧ 𝑦 ∈ (ℤ≥‘0)) → 𝑦 ∈ ℕ0)
59	52, 57, 58	sylancr 695	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((0 ≤ 𝑦 ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℕ0)
60	59	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (0 ≤ 𝑦 → (𝑦 ∈ ℤ → 𝑦 ∈ ℕ0))
61	51, 60	syl6com 37	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((0 ≤ 𝐿 ∧ 𝐿 ≤ 𝑦) → (𝑦 ∈ ℤ → (𝑦 ∈ ℤ → 𝑦 ∈ ℕ0)))
62	61	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (0 ≤ 𝐿 → (𝐿 ≤ 𝑦 → (𝑦 ∈ ℤ → (𝑦 ∈ ℤ → 𝑦 ∈ ℕ0))))
63	62	com14 96	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ℤ → (𝐿 ≤ 𝑦 → (𝑦 ∈ ℤ → (0 ≤ 𝐿 → 𝑦 ∈ ℕ0))))
64	63	pm2.43a 54	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℤ → (𝐿 ≤ 𝑦 → (0 ≤ 𝐿 → 𝑦 ∈ ℕ0)))
65	64	imp 445	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 ∈ ℤ ∧ 𝐿 ≤ 𝑦) → (0 ≤ 𝐿 → 𝑦 ∈ ℕ0))
66	65	com12 32	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0 ≤ 𝐿 → ((𝑦 ∈ ℤ ∧ 𝐿 ≤ 𝑦) → 𝑦 ∈ ℕ0))
67	3, 45, 66	mp2b 10	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ ℤ ∧ 𝐿 ≤ 𝑦) → 𝑦 ∈ ℕ0)
68	67	3adant1 1079	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿 ≤ 𝑦) → 𝑦 ∈ ℕ0)
69		eqcom 2629	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 + 1) = (#‘𝑣) ↔ (#‘𝑣) = (𝑦 + 1))
70		nn0p1gt0 11322	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ℕ0 → 0 < (𝑦 + 1))
71	70	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ ℕ0 ∧ (#‘𝑣) = (𝑦 + 1)) → 0 < (𝑦 + 1))
72		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ ℕ0 ∧ (#‘𝑣) = (𝑦 + 1)) → (#‘𝑣) = (𝑦 + 1))
73	71, 72	breqtrrd 4681	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 ∈ ℕ0 ∧ (#‘𝑣) = (𝑦 + 1)) → 0 < (#‘𝑣))
74	69, 73	sylan2b 492	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ ℕ0 ∧ (𝑦 + 1) = (#‘𝑣)) → 0 < (#‘𝑣))
75	74	adantrl 752	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣))) → 0 < (#‘𝑣))
76		vex 3203	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑣 ∈ V
77		hashgt0elex 13189	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑣 ∈ V ∧ 0 < (#‘𝑣)) → ∃𝑛 𝑛 ∈ 𝑣)
78		fi1uzindOLD.3	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 ∈ 𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌)
79	76	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → 𝑣 ∈ V)
80		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → 𝑛 ∈ 𝑣)
81		simpl 473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → 𝑦 ∈ ℕ0)
82		brfi1indlem 13278	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑣 ∈ V ∧ 𝑛 ∈ 𝑣 ∧ 𝑦 ∈ ℕ0) → ((#‘𝑣) = (𝑦 + 1) → (#‘(𝑣 ∖ {𝑛})) = 𝑦))
83	69, 82	syl5bi 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑣 ∈ V ∧ 𝑛 ∈ 𝑣 ∧ 𝑦 ∈ ℕ0) → ((𝑦 + 1) = (#‘𝑣) → (#‘(𝑣 ∖ {𝑛})) = 𝑦))
84	79, 80, 81, 83	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → ((𝑦 + 1) = (#‘𝑣) → (#‘(𝑣 ∖ {𝑛})) = 𝑦))
85	84	imp 445	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (𝑦 + 1) = (#‘𝑣)) → (#‘(𝑣 ∖ {𝑛})) = 𝑦)
86		peano2nn0 11333	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ0)
87	86	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (𝑦 + 1) = (#‘𝑣)) → (𝑦 + 1) ∈ ℕ0)
88	87	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((((∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (𝑦 + 1) = (#‘𝑣))) ∧ [𝑣 / 𝑎][𝑒 / 𝑏]𝜌) → (𝑦 + 1) ∈ ℕ0)
89		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((((∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (𝑦 + 1) = (#‘𝑣))) ∧ [𝑣 / 𝑎][𝑒 / 𝑏]𝜌) → [𝑣 / 𝑎][𝑒 / 𝑏]𝜌)
90		simplrr 801	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((((∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (𝑦 + 1) = (#‘𝑣))) ∧ [𝑣 / 𝑎][𝑒 / 𝑏]𝜌) → (𝑦 + 1) = (#‘𝑣))
91		simprlr 803	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (((∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (𝑦 + 1) = (#‘𝑣))) → 𝑛 ∈ 𝑣)
92	91	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ ((((∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (𝑦 + 1) = (#‘𝑣))) ∧ [𝑣 / 𝑎][𝑒 / 𝑏]𝜌) → 𝑛 ∈ 𝑣)
93	89, 90, 92	3jca 1242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((((∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (𝑦 + 1) = (#‘𝑣))) ∧ [𝑣 / 𝑎][𝑒 / 𝑏]𝜌) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣) ∧ 𝑛 ∈ 𝑣))
94	88, 93	jca 554	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((((∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (𝑦 + 1) = (#‘𝑣))) ∧ [𝑣 / 𝑎][𝑒 / 𝑏]𝜌) → ((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣) ∧ 𝑛 ∈ 𝑣)))
95		difexg 4808	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ (𝑣 ∈ V → (𝑣 ∖ {𝑛}) ∈ V)
96	76, 95	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑣 ∖ {𝑛}) ∈ V
97		fi1uzindOLD.f	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ 𝐹 ∈ 𝑈
98		simpl 473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → 𝑤 = (𝑣 ∖ {𝑛}))
99		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
100	99	sbceq1d 3440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ([𝑓 / 𝑏]𝜌 ↔ [𝐹 / 𝑏]𝜌))
101	98, 100	sbceqbid 3442	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ↔ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌))
102		eqcom 2629	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ (𝑦 = (#‘𝑤) ↔ (#‘𝑤) = 𝑦)
103		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ⊢ (𝑤 = (𝑣 ∖ {𝑛}) → (#‘𝑤) = (#‘(𝑣 ∖ {𝑛})))
104	103	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ (𝑤 = (𝑣 ∖ {𝑛}) → ((#‘𝑤) = 𝑦 ↔ (#‘(𝑣 ∖ {𝑛})) = 𝑦))
105	102, 104	syl5bb 272	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (𝑤 = (𝑣 ∖ {𝑛}) → (𝑦 = (#‘𝑤) ↔ (#‘(𝑣 ∖ {𝑛})) = 𝑦))
106	105	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝑦 = (#‘𝑤) ↔ (#‘(𝑣 ∖ {𝑛})) = 𝑦))
107	101, 106	anbi12d 747	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) ↔ ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦)))
108		fi1uzindOLD.4	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒))
109	107, 108	imbi12d 334	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ((([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) ↔ (([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
110	109	spc2gv 3296	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (((𝑣 ∖ {𝑛}) ∈ V ∧ 𝐹 ∈ 𝑈) → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → (([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
111	96, 97, 110	mp2an 708	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → (([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒))
112	111	expdimp 453	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) → ((#‘(𝑣 ∖ {𝑛})) = 𝑦 → 𝜒))
113	112	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((((∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (𝑦 + 1) = (#‘𝑣))) ∧ [𝑣 / 𝑎][𝑒 / 𝑏]𝜌) → ((#‘(𝑣 ∖ {𝑛})) = 𝑦 → 𝜒))
114	69	3anbi2i 1254	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣) ∧ 𝑛 ∈ 𝑣) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣))
115	114	anbi2i 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣) ∧ 𝑛 ∈ 𝑣)) ↔ ((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)))
116		fi1uzindOLD.step	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)
117	115, 116	sylanb 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)
118	94, 113, 117	syl6an 568	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((((∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (𝑦 + 1) = (#‘𝑣))) ∧ [𝑣 / 𝑎][𝑒 / 𝑏]𝜌) → ((#‘(𝑣 ∖ {𝑛})) = 𝑦 → 𝜓))
119	118	exp41 638	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 → (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (𝑦 + 1) = (#‘𝑣)) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → ((#‘(𝑣 ∖ {𝑛})) = 𝑦 → 𝜓)))))
120	119	com15 101	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((#‘(𝑣 ∖ {𝑛})) = 𝑦 → ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 → (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (𝑦 + 1) = (#‘𝑣)) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
121	120	com23 86	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((#‘(𝑣 ∖ {𝑛})) = 𝑦 → (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (𝑦 + 1) = (#‘𝑣)) → ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
122	85, 121	mpcom 38	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (𝑦 + 1) = (#‘𝑣)) → ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))))
123	122	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → ((𝑦 + 1) = (#‘𝑣) → ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
124	123	com23 86	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 → ((𝑦 + 1) = (#‘𝑣) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
125	124	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑦 ∈ ℕ0 → (𝑛 ∈ 𝑣 → ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 → ((𝑦 + 1) = (#‘𝑣) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))))))
126	125	com15 101	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (𝑛 ∈ 𝑣 → ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))))))
127	126	imp 445	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 ∈ 𝑣) → ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
128	78, 127	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 ∈ 𝑣) → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))))
129	128	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (𝑛 ∈ 𝑣 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
130	129	com4l 92	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 ∈ 𝑣 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
131	130	exlimiv 1858	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (∃𝑛 𝑛 ∈ 𝑣 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
132	77, 131	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑣 ∈ V ∧ 0 < (#‘𝑣)) → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
133	132	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑣 ∈ V → (0 < (#‘𝑣) → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))))))
134	133	com25 99	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 ∈ V → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → (0 < (#‘𝑣) → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))))))
135	76, 134	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → (0 < (#‘𝑣) → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
136	135	imp 445	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → (𝑦 ∈ ℕ0 → (0 < (#‘𝑣) → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))))
137	136	impcom 446	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣))) → (0 < (#‘𝑣) → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))
138	75, 137	mpd 15	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣))) → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))
139	68, 138	sylan 488	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿 ≤ 𝑦) ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣))) → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))
140	139	impancom 456	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿 ≤ 𝑦) ∧ ∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃)) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓))
141	140	alrimivv 1856	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿 ≤ 𝑦) ∧ ∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃)) → ∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓))
142	141	ex 450	. . . . . . . . . . . . . . . 16 ⊢ ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿 ≤ 𝑦) → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → ∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓)))
143	44, 142	syl5bi 232	. . . . . . . . . . . . . . 15 ⊢ ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿 ≤ 𝑦) → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑣)) → 𝜓) → ∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓)))
144	16, 20, 24, 28, 33, 143	uzind 11469	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝐿 ≤ 𝑛) → ∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑣)) → 𝜓))
145	5, 7, 12, 144	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ (((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (#‘𝑉)) → ∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑣)) → 𝜓))
146		sbcex 3445	. . . . . . . . . . . . . . . 16 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → 𝑉 ∈ V)
147		sbccom 3509	. . . . . . . . . . . . . . . . 17 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ↔ [𝐸 / 𝑏][𝑉 / 𝑎]𝜌)
148		sbcex 3445	. . . . . . . . . . . . . . . . 17 ⊢ ([𝐸 / 𝑏][𝑉 / 𝑎]𝜌 → 𝐸 ∈ V)
149	147, 148	sylbi 207	. . . . . . . . . . . . . . . 16 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → 𝐸 ∈ V)
150	146, 149	jca 554	. . . . . . . . . . . . . . 15 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
151		simpl 473	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → 𝑣 = 𝑉)
152		simpr 477	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → 𝑒 = 𝐸)
153	152	sbceq1d 3440	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ([𝑒 / 𝑏]𝜌 ↔ [𝐸 / 𝑏]𝜌))
154	151, 153	sbceqbid 3442	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ↔ [𝑉 / 𝑎][𝐸 / 𝑏]𝜌))
155		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 = 𝑉 → (#‘𝑣) = (#‘𝑉))
156	155	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = 𝑉 → (𝑛 = (#‘𝑣) ↔ 𝑛 = (#‘𝑉)))
157	156	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑛 = (#‘𝑣) ↔ 𝑛 = (#‘𝑉)))
158	154, 157	anbi12d 747	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑣)) ↔ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑉))))
159		fi1uzindOLD.1	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑))
160	158, 159	imbi12d 334	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑣)) → 𝜓) ↔ (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑉)) → 𝜑)))
161	160	spc2gv 3296	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑣)) → 𝜓) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑉)) → 𝜑)))
162	161	com23 86	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑉)) → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑣)) → 𝜓) → 𝜑)))
163	162	expd 452	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑛 = (#‘𝑉) → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑣)) → 𝜓) → 𝜑))))
164	150, 163	mpcom 38	. . . . . . . . . . . . . 14 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑛 = (#‘𝑉) → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑣)) → 𝜓) → 𝜑)))
165	164	imp 445	. . . . . . . . . . . . 13 ⊢ (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑉)) → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑣)) → 𝜓) → 𝜑))
166	145, 165	syl5com 31	. . . . . . . . . . . 12 ⊢ (((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (#‘𝑉)) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑉)) → 𝜑))
167	166	exp31 630	. . . . . . . . . . 11 ⊢ (𝐿 ≤ (#‘𝑉) → (𝑛 ∈ ℕ0 → (𝑛 = (#‘𝑉) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑉)) → 𝜑))))
168	167	com14 96	. . . . . . . . . 10 ⊢ (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑉)) → (𝑛 ∈ ℕ0 → (𝑛 = (#‘𝑉) → (𝐿 ≤ (#‘𝑉) → 𝜑))))
169	168	expcom 451	. . . . . . . . 9 ⊢ (𝑛 = (#‘𝑉) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑛 ∈ ℕ0 → (𝑛 = (#‘𝑉) → (𝐿 ≤ (#‘𝑉) → 𝜑)))))
170	169	com24 95	. . . . . . . 8 ⊢ (𝑛 = (#‘𝑉) → (𝑛 = (#‘𝑉) → (𝑛 ∈ ℕ0 → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑)))))
171	170	pm2.43i 52	. . . . . . 7 ⊢ (𝑛 = (#‘𝑉) → (𝑛 ∈ ℕ0 → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑))))
172	171	imp 445	. . . . . 6 ⊢ ((𝑛 = (#‘𝑉) ∧ 𝑛 ∈ ℕ0) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑)))
173	172	exlimiv 1858	. . . . 5 ⊢ (∃𝑛(𝑛 = (#‘𝑉) ∧ 𝑛 ∈ ℕ0) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑)))
174	2, 173	sylbi 207	. . . 4 ⊢ ((#‘𝑉) ∈ ℕ0 → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑)))
175	1, 174	syl 17	. . 3 ⊢ (𝑉 ∈ Fin → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑)))
176	175	com12 32	. 2 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑉 ∈ Fin → (𝐿 ≤ (#‘𝑉) → 𝜑)))
177	176	3imp 1256	1 ⊢ (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Vcvv 3200  [wsbc 3435   ∖ cdif 3571  {csn 4177   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  Fincfn 7955  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   < clt 10074   ≤ cle 10075  ℕ₀cn0 11292  ℤcz 11377  ℤ_≥cuz 11687  #chash 13117

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-oadd 7564  df-er 7742  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-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118

This theorem is referenced by:  brfi1uzindOLD  13286  opfi1uzindOLD  13289