Theorem fi1uzind 13317

Description: Properties of an ordered pair with a finite first component with at least L elements, proven by finite induction on the size of the first component. This theorem can be applied for graphs (represented as orderd pairs of vertices and edges) with a finite number of vertices, usually with 𝐿 = 0 (see opfi1ind 13322) or 𝐿 = 1. (Contributed by AV, 22-Oct-2020.) (Revised by AV, 28-Mar-2021.)

Hypotheses

Ref	Expression
fi1uzind.f	⊢ 𝐹 ∈ V
fi1uzind.l	⊢ 𝐿 ∈ ℕ0
fi1uzind.1	⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑))
fi1uzind.2	⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃))
fi1uzind.3	⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 ∈ 𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌)
fi1uzind.4	⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒))
fi1uzind.base	⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = 𝐿) → 𝜓)
fi1uzind.step	⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)

Assertion

Ref	Expression
fi1uzind	⊢ (([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)   𝐸(𝑦,𝑤,𝑓,𝑏)   𝐹(𝑦,𝑣,𝑒,𝑛,𝑏)   𝐿(𝑤,𝑓,𝑎,𝑏)   𝑉(𝑦,𝑤,𝑓)

Proof of Theorem fi1uzind

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clel 2647	. . . 4 ⊢ ((#‘𝑉) ∈ ℕ0 ↔ ∃𝑛(𝑛 = (#‘𝑉) ∧ 𝑛 ∈ ℕ0))
2		fi1uzind.l	. . . . . . . . . . . . . 14 ⊢ 𝐿 ∈ ℕ0
3		nn0z 11438	. . . . . . . . . . . . . 14 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℤ)
4	2, 3	mp1i 13	. . . . . . . . . . . . 13 ⊢ (((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (#‘𝑉)) → 𝐿 ∈ ℤ)
5		nn0z 11438	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℤ)
6	5	ad2antlr 763	. . . . . . . . . . . . 13 ⊢ (((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (#‘𝑉)) → 𝑛 ∈ ℤ)
7		breq2 4689	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝑉) = 𝑛 → (𝐿 ≤ (#‘𝑉) ↔ 𝐿 ≤ 𝑛))
8	7	eqcoms 2659	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (#‘𝑉) → (𝐿 ≤ (#‘𝑉) ↔ 𝐿 ≤ 𝑛))
9	8	biimpcd 239	. . . . . . . . . . . . . . 15 ⊢ (𝐿 ≤ (#‘𝑉) → (𝑛 = (#‘𝑉) → 𝐿 ≤ 𝑛))
10	9	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) → (𝑛 = (#‘𝑉) → 𝐿 ≤ 𝑛))
11	10	imp 444	. . . . . . . . . . . . 13 ⊢ (((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (#‘𝑉)) → 𝐿 ≤ 𝑛)
12		eqeq1 2655	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐿 → (𝑥 = (#‘𝑣) ↔ 𝐿 = (#‘𝑣)))
13	12	anbi2d 740	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐿 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (#‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝐿 = (#‘𝑣))))
14	13	imbi1d 330	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐿 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (#‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝐿 = (#‘𝑣)) → 𝜓)))
15	14	2albidv 1891	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐿 → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (#‘𝑣)) → 𝜓) ↔ ∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝐿 = (#‘𝑣)) → 𝜓)))
16		eqeq1 2655	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝑥 = (#‘𝑣) ↔ 𝑦 = (#‘𝑣)))
17	16	anbi2d 740	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (#‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑣))))
18	17	imbi1d 330	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (#‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑣)) → 𝜓)))
19	18	2albidv 1891	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (#‘𝑣)) → 𝜓) ↔ ∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑣)) → 𝜓)))
20		eqeq1 2655	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 = (#‘𝑣) ↔ (𝑦 + 1) = (#‘𝑣)))
21	20	anbi2d 740	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 + 1) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (#‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣))))
22	21	imbi1d 330	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑦 + 1) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (#‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓)))
23	22	2albidv 1891	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 + 1) → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (#‘𝑣)) → 𝜓) ↔ ∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣)) → 𝜓)))
24		eqeq1 2655	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑛 → (𝑥 = (#‘𝑣) ↔ 𝑛 = (#‘𝑣)))
25	24	anbi2d 740	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑛 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (#‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑣))))
26	25	imbi1d 330	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑛 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (#‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑣)) → 𝜓)))
27	26	2albidv 1891	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑛 → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (#‘𝑣)) → 𝜓) ↔ ∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑣)) → 𝜓)))
28		eqcom 2658	. . . . . . . . . . . . . . . . 17 ⊢ (𝐿 = (#‘𝑣) ↔ (#‘𝑣) = 𝐿)
29		fi1uzind.base	. . . . . . . . . . . . . . . . 17 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = 𝐿) → 𝜓)
30	28, 29	sylan2b 491	. . . . . . . . . . . . . . . 16 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝐿 = (#‘𝑣)) → 𝜓)
31	30	gen2 1763	. . . . . . . . . . . . . . 15 ⊢ ∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝐿 = (#‘𝑣)) → 𝜓)
32	31	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐿 ∈ ℤ → ∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝐿 = (#‘𝑣)) → 𝜓))
33		simpl 472	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → 𝑣 = 𝑤)
34		simpr 476	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → 𝑒 = 𝑓)
35	34	sbceq1d 3473	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ([𝑒 / 𝑏]𝜌 ↔ [𝑓 / 𝑏]𝜌))
36	33, 35	sbceqbid 3475	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ↔ [𝑤 / 𝑎][𝑓 / 𝑏]𝜌))
37		fveq2 6229	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝑤 → (#‘𝑣) = (#‘𝑤))
38	37	eqeq2d 2661	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝑤 → (𝑦 = (#‘𝑣) ↔ 𝑦 = (#‘𝑤)))
39	38	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝑦 = (#‘𝑣) ↔ 𝑦 = (#‘𝑤)))
40	36, 39	anbi12d 747	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑣)) ↔ ([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤))))
41		fi1uzind.2	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃))
42	40, 41	imbi12d 333	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑣)) → 𝜓) ↔ (([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃)))
43	42	cbval2v 2321	. . . . . . . . . . . . . . 15 ⊢ (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑣)) → 𝜓) ↔ ∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃))
44		nn0ge0 11356	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐿 ∈ ℕ0 → 0 ≤ 𝐿)
45		0red 10079	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ℤ → 0 ∈ ℝ)
46		nn0re 11339	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℝ)
47	2, 46	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ℤ → 𝐿 ∈ ℝ)
48		zre 11419	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℝ)
49		letr 10169	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((0 ∈ ℝ ∧ 𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((0 ≤ 𝐿 ∧ 𝐿 ≤ 𝑦) → 0 ≤ 𝑦))
50	45, 47, 48, 49	syl3anc 1366	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ ℤ → ((0 ≤ 𝐿 ∧ 𝐿 ≤ 𝑦) → 0 ≤ 𝑦))
51		0nn0 11345	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 0 ∈ ℕ0
52		pm3.22 464	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((0 ≤ 𝑦 ∧ 𝑦 ∈ ℤ) → (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦))
53		0z 11426	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 0 ∈ ℤ
54		eluz1 11729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (0 ∈ ℤ → (𝑦 ∈ (ℤ≥‘0) ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦)))
55	53, 54	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((0 ≤ 𝑦 ∧ 𝑦 ∈ ℤ) → (𝑦 ∈ (ℤ≥‘0) ↔ (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦)))
56	52, 55	mpbird 247	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((0 ≤ 𝑦 ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ (ℤ≥‘0))
57		eluznn0 11795	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((0 ∈ ℕ0 ∧ 𝑦 ∈ (ℤ≥‘0)) → 𝑦 ∈ ℕ0)
58	51, 56, 57	sylancr 696	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((0 ≤ 𝑦 ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℕ0)
59	58	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (0 ≤ 𝑦 → (𝑦 ∈ ℤ → 𝑦 ∈ ℕ0))
60	50, 59	syl6com 37	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((0 ≤ 𝐿 ∧ 𝐿 ≤ 𝑦) → (𝑦 ∈ ℤ → (𝑦 ∈ ℤ → 𝑦 ∈ ℕ0)))
61	60	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0 ≤ 𝐿 → (𝐿 ≤ 𝑦 → (𝑦 ∈ ℤ → (𝑦 ∈ ℤ → 𝑦 ∈ ℕ0))))
62	61	com14 96	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℤ → (𝐿 ≤ 𝑦 → (𝑦 ∈ ℤ → (0 ≤ 𝐿 → 𝑦 ∈ ℕ0))))
63	62	pm2.43a 54	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℤ → (𝐿 ≤ 𝑦 → (0 ≤ 𝐿 → 𝑦 ∈ ℕ0)))
64	63	imp 444	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ ℤ ∧ 𝐿 ≤ 𝑦) → (0 ≤ 𝐿 → 𝑦 ∈ ℕ0))
65	64	com12 32	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0 ≤ 𝐿 → ((𝑦 ∈ ℤ ∧ 𝐿 ≤ 𝑦) → 𝑦 ∈ ℕ0))
66	2, 44, 65	mp2b 10	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℤ ∧ 𝐿 ≤ 𝑦) → 𝑦 ∈ ℕ0)
67	66	3adant1 1099	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿 ≤ 𝑦) → 𝑦 ∈ ℕ0)
68		eqcom 2658	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 + 1) = (#‘𝑣) ↔ (#‘𝑣) = (𝑦 + 1))
69		nn0p1gt0 11360	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℕ0 → 0 < (𝑦 + 1))
70	69	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 ∈ ℕ0 ∧ (#‘𝑣) = (𝑦 + 1)) → 0 < (𝑦 + 1))
71		simpr 476	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 ∈ ℕ0 ∧ (#‘𝑣) = (𝑦 + 1)) → (#‘𝑣) = (𝑦 + 1))
72	70, 71	breqtrrd 4713	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ ℕ0 ∧ (#‘𝑣) = (𝑦 + 1)) → 0 < (#‘𝑣))
73	68, 72	sylan2b 491	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ ℕ0 ∧ (𝑦 + 1) = (#‘𝑣)) → 0 < (#‘𝑣))
74	73	adantrl 752	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣))) → 0 < (#‘𝑣))
75		vex 3234	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑣 ∈ V
76		hashgt0elex 13227	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑣 ∈ V ∧ 0 < (#‘𝑣)) → ∃𝑛 𝑛 ∈ 𝑣)
77		fi1uzind.3	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 ∈ 𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌)
78	75	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → 𝑣 ∈ V)
79		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → 𝑛 ∈ 𝑣)
80		simpl 472	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → 𝑦 ∈ ℕ0)
81		brfi1indlem 13316	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑣 ∈ V ∧ 𝑛 ∈ 𝑣 ∧ 𝑦 ∈ ℕ0) → ((#‘𝑣) = (𝑦 + 1) → (#‘(𝑣 ∖ {𝑛})) = 𝑦))
82	68, 81	syl5bi 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑣 ∈ V ∧ 𝑛 ∈ 𝑣 ∧ 𝑦 ∈ ℕ0) → ((𝑦 + 1) = (#‘𝑣) → (#‘(𝑣 ∖ {𝑛})) = 𝑦))
83	78, 79, 80, 82	syl3anc 1366	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → ((𝑦 + 1) = (#‘𝑣) → (#‘(𝑣 ∖ {𝑛})) = 𝑦))
84	83	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (𝑦 + 1) = (#‘𝑣)) → (#‘(𝑣 ∖ {𝑛})) = 𝑦)
85		peano2nn0 11371	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ0)
86	85	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (𝑦 + 1) = (#‘𝑣)) → (𝑦 + 1) ∈ ℕ0)
87	86	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
88		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
89		simplrr 818	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) = (#‘𝑣))
90		simprlr 820	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((∀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)
91	90	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) → 𝑛 ∈ 𝑣)
92	88, 89, 91	3jca 1261	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣) ∧ 𝑛 ∈ 𝑣))
93	87, 92	jca 553	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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) → ((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣) ∧ 𝑛 ∈ 𝑣)))
94		difexg 4841	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑣 ∈ V → (𝑣 ∖ {𝑛}) ∈ V)
95	75, 94	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑣 ∖ {𝑛}) ∈ V
96		fi1uzind.f	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ 𝐹 ∈ V
97		simpl 472	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → 𝑤 = (𝑣 ∖ {𝑛}))
98		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
99	98	sbceq1d 3473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ([𝑓 / 𝑏]𝜌 ↔ [𝐹 / 𝑏]𝜌))
100	97, 99	sbceqbid 3475	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ↔ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌))
101		eqcom 2658	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (𝑦 = (#‘𝑤) ↔ (#‘𝑤) = 𝑦)
102		fveq2 6229	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 ⊢ (𝑤 = (𝑣 ∖ {𝑛}) → (#‘𝑤) = (#‘(𝑣 ∖ {𝑛})))
103	102	eqeq1d 2653	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (𝑤 = (𝑣 ∖ {𝑛}) → ((#‘𝑤) = 𝑦 ↔ (#‘(𝑣 ∖ {𝑛})) = 𝑦))
104	101, 103	syl5bb 272	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (𝑤 = (𝑣 ∖ {𝑛}) → (𝑦 = (#‘𝑤) ↔ (#‘(𝑣 ∖ {𝑛})) = 𝑦))
105	104	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝑦 = (#‘𝑤) ↔ (#‘(𝑣 ∖ {𝑛})) = 𝑦))
106	100, 105	anbi12d 747	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) ↔ ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦)))
107		fi1uzind.4	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒))
108	106, 107	imbi12d 333	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ((([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) ↔ (([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
109	108	spc2gv 3327	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (((𝑣 ∖ {𝑛}) ∈ V ∧ 𝐹 ∈ V) → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → (([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
110	95, 96, 109	mp2an 708	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → (([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒))
111	110	expdimp 452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) → ((#‘(𝑣 ∖ {𝑛})) = 𝑦 → 𝜒))
112	111	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 → 𝜒))
113	68	3anbi2i 1273	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣) ∧ 𝑛 ∈ 𝑣) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣))
114	113	anbi2i 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ (((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)))
115		fi1uzind.step	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)
116	114, 115	sylanb 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)
117	93, 112, 116	syl6an 567	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
118	117	exp41 637	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (∀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	com15 101	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((#‘(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)))))
120	119	com23 86	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((#‘(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)))))
121	84, 120	mpcom 38	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
122	121	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((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	com23 86	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
124	123	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (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	com15 101	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))))
126	125	imp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (([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	77, 126	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 ∈ 𝑣) → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))))
128	127	ex 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ([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	com4l 92	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 ∈ 𝑣 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
130	129	exlimiv 1898	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (∃𝑛 𝑛 ∈ 𝑣 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
131	76, 130	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 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)))))
132	131	ex 449	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (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	com25 99	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (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))))))
134	75, 133	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → ((𝑦 + 1) = (#‘𝑣) → (𝑦 ∈ ℕ0 → (0 < (#‘𝑣) → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))))
135	134	imp 444	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (([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	impcom 445	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣))) → (0 < (#‘𝑣) → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓)))
137	74, 136	mpd 15	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (#‘𝑣))) → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (#‘𝑤)) → 𝜃) → 𝜓))
138	67, 137	sylan 487	. . . . . . . . . . . . . . . . . 18 ⊢ (((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))
139	138	impancom 455	. . . . . . . . . . . . . . . . 17 ⊢ (((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))
140	139	alrimivv 1896	. . . . . . . . . . . . . . . 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))
141	140	ex 449	. . . . . . . . . . . . . . 15 ⊢ ((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	43, 141	syl5bi 232	. . . . . . . . . . . . . 14 ⊢ ((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)))
143	15, 19, 23, 27, 32, 142	uzind 11507	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝐿 ≤ 𝑛) → ∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑣)) → 𝜓))
144	4, 6, 11, 143	syl3anc 1366	. . . . . . . . . . . 12 ⊢ (((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (#‘𝑉)) → ∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑣)) → 𝜓))
145		sbcex 3478	. . . . . . . . . . . . . . 15 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → 𝑉 ∈ V)
146		sbccom 3542	. . . . . . . . . . . . . . . 16 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ↔ [𝐸 / 𝑏][𝑉 / 𝑎]𝜌)
147		sbcex 3478	. . . . . . . . . . . . . . . 16 ⊢ ([𝐸 / 𝑏][𝑉 / 𝑎]𝜌 → 𝐸 ∈ V)
148	146, 147	sylbi 207	. . . . . . . . . . . . . . 15 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → 𝐸 ∈ V)
149	145, 148	jca 553	. . . . . . . . . . . . . 14 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
150		simpl 472	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → 𝑣 = 𝑉)
151		simpr 476	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → 𝑒 = 𝐸)
152	151	sbceq1d 3473	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ([𝑒 / 𝑏]𝜌 ↔ [𝐸 / 𝑏]𝜌))
153	150, 152	sbceqbid 3475	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ↔ [𝑉 / 𝑎][𝐸 / 𝑏]𝜌))
154		fveq2 6229	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = 𝑉 → (#‘𝑣) = (#‘𝑉))
155	154	eqeq2d 2661	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝑉 → (𝑛 = (#‘𝑣) ↔ 𝑛 = (#‘𝑉)))
156	155	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑛 = (#‘𝑣) ↔ 𝑛 = (#‘𝑉)))
157	153, 156	anbi12d 747	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑣)) ↔ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑉))))
158		fi1uzind.1	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑))
159	157, 158	imbi12d 333	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑣)) → 𝜓) ↔ (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑉)) → 𝜑)))
160	159	spc2gv 3327	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑣)) → 𝜓) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑉)) → 𝜑)))
161	160	com23 86	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑉)) → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑣)) → 𝜓) → 𝜑)))
162	161	expd 451	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑛 = (#‘𝑉) → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑣)) → 𝜓) → 𝜑))))
163	149, 162	mpcom 38	. . . . . . . . . . . . 13 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑛 = (#‘𝑉) → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑣)) → 𝜓) → 𝜑)))
164	163	imp 444	. . . . . . . . . . . 12 ⊢ (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑉)) → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑣)) → 𝜓) → 𝜑))
165	144, 164	syl5com 31	. . . . . . . . . . 11 ⊢ (((𝐿 ≤ (#‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (#‘𝑉)) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑉)) → 𝜑))
166	165	exp31 629	. . . . . . . . . 10 ⊢ (𝐿 ≤ (#‘𝑉) → (𝑛 ∈ ℕ0 → (𝑛 = (#‘𝑉) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑉)) → 𝜑))))
167	166	com14 96	. . . . . . . . 9 ⊢ (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (#‘𝑉)) → (𝑛 ∈ ℕ0 → (𝑛 = (#‘𝑉) → (𝐿 ≤ (#‘𝑉) → 𝜑))))
168	167	expcom 450	. . . . . . . 8 ⊢ (𝑛 = (#‘𝑉) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑛 ∈ ℕ0 → (𝑛 = (#‘𝑉) → (𝐿 ≤ (#‘𝑉) → 𝜑)))))
169	168	com24 95	. . . . . . 7 ⊢ (𝑛 = (#‘𝑉) → (𝑛 = (#‘𝑉) → (𝑛 ∈ ℕ0 → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑)))))
170	169	pm2.43i 52	. . . . . 6 ⊢ (𝑛 = (#‘𝑉) → (𝑛 ∈ ℕ0 → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑))))
171	170	imp 444	. . . . 5 ⊢ ((𝑛 = (#‘𝑉) ∧ 𝑛 ∈ ℕ0) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑)))
172	171	exlimiv 1898	. . . 4 ⊢ (∃𝑛(𝑛 = (#‘𝑉) ∧ 𝑛 ∈ ℕ0) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑)))
173	1, 172	sylbi 207	. . 3 ⊢ ((#‘𝑉) ∈ ℕ0 → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (#‘𝑉) → 𝜑)))
174		hashcl 13185	. . 3 ⊢ (𝑉 ∈ Fin → (#‘𝑉) ∈ ℕ0)
175	173, 174	syl11 33	. 2 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑉 ∈ Fin → (𝐿 ≤ (#‘𝑉) → 𝜑)))
176	175	3imp 1275	1 ⊢ (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054  ∀wal 1521   = wceq 1523  ∃wex 1744   ∈ wcel 2030  Vcvv 3231  [wsbc 3468   ∖ cdif 3604  {csn 4210   class class class wbr 4685  ‘cfv 5926  (class class class)co 6690  Fincfn 7997  ℝcr 9973  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112   ≤ cle 10113  ℕ₀cn0 11330  ℤcz 11415  ℤ_≥cuz 11725  #chash 13157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-hash 13158

This theorem is referenced by:  brfi1uzind  13318  opfi1uzind  13321