Theorem fi1uzind 14063

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 14068) 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		dfclel 2817	. . . 4 ⊢ ((♯‘𝑉) ∈ ℕ0 ↔ ∃𝑛(𝑛 = (♯‘𝑉) ∧ 𝑛 ∈ ℕ0))
2		fi1uzind.l	. . . . . . . . . . . . . 14 ⊢ 𝐿 ∈ ℕ0
3		nn0z 12200	. . . . . . . . . . . . . 14 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℤ)
4	2, 3	mp1i 13	. . . . . . . . . . . . 13 ⊢ (((𝐿 ≤ (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (♯‘𝑉)) → 𝐿 ∈ ℤ)
5		nn0z 12200	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℤ)
6	5	ad2antlr 727	. . . . . . . . . . . . 13 ⊢ (((𝐿 ≤ (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (♯‘𝑉)) → 𝑛 ∈ ℤ)
7		breq2 5057	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑉) = 𝑛 → (𝐿 ≤ (♯‘𝑉) ↔ 𝐿 ≤ 𝑛))
8	7	eqcoms 2745	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (♯‘𝑉) → (𝐿 ≤ (♯‘𝑉) ↔ 𝐿 ≤ 𝑛))
9	8	biimpcd 252	. . . . . . . . . . . . . . 15 ⊢ (𝐿 ≤ (♯‘𝑉) → (𝑛 = (♯‘𝑉) → 𝐿 ≤ 𝑛))
10	9	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ≤ (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) → (𝑛 = (♯‘𝑉) → 𝐿 ≤ 𝑛))
11	10	imp 410	. . . . . . . . . . . . 13 ⊢ (((𝐿 ≤ (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (♯‘𝑉)) → 𝐿 ≤ 𝑛)
12		eqeq1 2741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐿 → (𝑥 = (♯‘𝑣) ↔ 𝐿 = (♯‘𝑣)))
13	12	anbi2d 632	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐿 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (♯‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝐿 = (♯‘𝑣))))
14	13	imbi1d 345	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐿 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (♯‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝐿 = (♯‘𝑣)) → 𝜓)))
15	14	2albidv 1931	. . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝑥 = (♯‘𝑣) ↔ 𝑦 = (♯‘𝑣)))
17	16	anbi2d 632	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (♯‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑣))))
18	17	imbi1d 345	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (♯‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑣)) → 𝜓)))
19	18	2albidv 1931	. . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 = (♯‘𝑣) ↔ (𝑦 + 1) = (♯‘𝑣)))
21	20	anbi2d 632	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 + 1) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (♯‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣))))
22	21	imbi1d 345	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑦 + 1) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (♯‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣)) → 𝜓)))
23	22	2albidv 1931	. . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑛 → (𝑥 = (♯‘𝑣) ↔ 𝑛 = (♯‘𝑣)))
25	24	anbi2d 632	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑛 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (♯‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑣))))
26	25	imbi1d 345	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑛 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (♯‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑣)) → 𝜓)))
27	26	2albidv 1931	. . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . . . . . 17 ⊢ (𝐿 = (♯‘𝑣) ↔ (♯‘𝑣) = 𝐿)
29		fi1uzind.base	. . . . . . . . . . . . . . . . 17 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (♯‘𝑣) = 𝐿) → 𝜓)
30	28, 29	sylan2b 597	. . . . . . . . . . . . . . . 16 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝐿 = (♯‘𝑣)) → 𝜓)
31	30	gen2 1804	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → 𝑣 = 𝑤)
34		simpr 488	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → 𝑒 = 𝑓)
35	34	sbceq1d 3699	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ([𝑒 / 𝑏]𝜌 ↔ [𝑓 / 𝑏]𝜌))
36	33, 35	sbceqbid 3701	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ↔ [𝑤 / 𝑎][𝑓 / 𝑏]𝜌))
37		fveq2 6717	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝑤 → (♯‘𝑣) = (♯‘𝑤))
38	37	eqeq2d 2748	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝑤 → (𝑦 = (♯‘𝑣) ↔ 𝑦 = (♯‘𝑤)))
39	38	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝑦 = (♯‘𝑣) ↔ 𝑦 = (♯‘𝑤)))
40	36, 39	anbi12d 634	. . . . . . . . . . . . . . . . 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 348	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑣)) → 𝜓) ↔ (([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑤)) → 𝜃)))
43	42	cbval2vw 2048	. . . . . . . . . . . . . . 15 ⊢ (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑣)) → 𝜓) ↔ ∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑤)) → 𝜃))
44		nn0ge0 12115	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐿 ∈ ℕ0 → 0 ≤ 𝐿)
45		0red 10836	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ℤ → 0 ∈ ℝ)
46		nn0re 12099	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℝ)
47	2, 46	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ℤ → 𝐿 ∈ ℝ)
48		zre 12180	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℝ)
49		letr 10926	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((0 ∈ ℝ ∧ 𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((0 ≤ 𝐿 ∧ 𝐿 ≤ 𝑦) → 0 ≤ 𝑦))
50	45, 47, 48, 49	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ ℤ → ((0 ≤ 𝐿 ∧ 𝐿 ≤ 𝑦) → 0 ≤ 𝑦))
51		0nn0 12105	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 0 ∈ ℕ0
52		pm3.22 463	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((0 ≤ 𝑦 ∧ 𝑦 ∈ ℤ) → (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦))
53		0z 12187	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 0 ∈ ℤ
54		eluz1 12442	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((0 ≤ 𝑦 ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ (ℤ≥‘0))
57		eluznn0 12513	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((0 ∈ ℕ0 ∧ 𝑦 ∈ (ℤ≥‘0)) → 𝑦 ∈ ℕ0)
58	51, 56, 57	sylancr 590	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((0 ≤ 𝑦 ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℕ0)
59	58	ex 416	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿 ≤ 𝑦) → 𝑦 ∈ ℕ0)
68		eqcom 2744	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 + 1) = (♯‘𝑣) ↔ (♯‘𝑣) = (𝑦 + 1))
69		nn0p1gt0 12119	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℕ0 → 0 < (𝑦 + 1))
70	69	adantr 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 ∈ ℕ0 ∧ (♯‘𝑣) = (𝑦 + 1)) → 0 < (𝑦 + 1))
71		simpr 488	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 ∈ ℕ0 ∧ (♯‘𝑣) = (𝑦 + 1)) → (♯‘𝑣) = (𝑦 + 1))
72	70, 71	breqtrrd 5081	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ ℕ0 ∧ (♯‘𝑣) = (𝑦 + 1)) → 0 < (♯‘𝑣))
73	68, 72	sylan2b 597	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ ℕ0 ∧ (𝑦 + 1) = (♯‘𝑣)) → 0 < (♯‘𝑣))
74	73	adantrl 716	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣))) → 0 < (♯‘𝑣))
75		hashgt0elex 13968	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑣 ∈ V ∧ 0 < (♯‘𝑣)) → ∃𝑛 𝑛 ∈ 𝑣)
76		fi1uzind.3	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 ∈ 𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌)
77		vex 3412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ 𝑣 ∈ V
78	77	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → 𝑣 ∈ V)
79		simpr 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → 𝑛 ∈ 𝑣)
80		simpl 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → 𝑦 ∈ ℕ0)
81		hashdifsnp1 14062	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑣 ∈ V ∧ 𝑛 ∈ 𝑣 ∧ 𝑦 ∈ ℕ0) → ((♯‘𝑣) = (𝑦 + 1) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
82	68, 81	syl5bi 245	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑣 ∈ V ∧ 𝑛 ∈ 𝑣 ∧ 𝑦 ∈ ℕ0) → ((𝑦 + 1) = (♯‘𝑣) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
83	78, 79, 80, 82	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → ((𝑦 + 1) = (♯‘𝑣) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
84	83	imp 410	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (𝑦 + 1) = (♯‘𝑣)) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦)
85		peano2nn0 12130	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ0)
86	85	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (𝑦 + 1) = (♯‘𝑣)) → (𝑦 + 1) ∈ ℕ0)
87	86	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1130	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 515	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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	77	difexi 5221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑣 ∖ {𝑛}) ∈ V
95		fi1uzind.f	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ 𝐹 ∈ V
96		simpl 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → 𝑤 = (𝑣 ∖ {𝑛}))
97		simpr 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
98	97	sbceq1d 3699	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ([𝑓 / 𝑏]𝜌 ↔ [𝐹 / 𝑏]𝜌))
99	96, 98	sbceqbid 3701	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ↔ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌))
100		eqcom 2744	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (𝑦 = (♯‘𝑤) ↔ (♯‘𝑤) = 𝑦)
101		fveqeq2 6726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (𝑤 = (𝑣 ∖ {𝑛}) → ((♯‘𝑤) = 𝑦 ↔ (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
102	100, 101	syl5bb 286	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (𝑤 = (𝑣 ∖ {𝑛}) → (𝑦 = (♯‘𝑤) ↔ (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
103	102	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝑦 = (♯‘𝑤) ↔ (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
104	99, 103	anbi12d 634	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑤)) ↔ ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (♯‘(𝑣 ∖ {𝑛})) = 𝑦)))
105		fi1uzind.4	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒))
106	104, 105	imbi12d 348	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ((([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑤)) → 𝜃) ↔ (([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (♯‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
107	106	spc2gv 3515	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
108	94, 95, 107	mp2an 692	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑤)) → 𝜃) → (([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (♯‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒))
109	108	expdimp 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) → ((♯‘(𝑣 ∖ {𝑛})) = 𝑦 → 𝜒))
110	109	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
111	68	3anbi2i 1160	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣) ∧ 𝑛 ∈ 𝑣) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣))
112	111	anbi2i 626	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
113		fi1uzind.step	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)
114	112, 113	sylanb 584	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)
115	93, 110, 114	syl6an 684	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
116	115	exp41 438	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
117	116	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)))))
118	117	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)))))
119	84, 118	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))))
120	119	ex 416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
121	120	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)))))
122	121	ex 416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))))
123	122	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))))))
124	123	imp 410	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
125	76, 124	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))))
126	125	ex 416	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → (𝑛 ∈ 𝑣 → ((𝑦 + 1) = (♯‘𝑣) → (𝑦 ∈ ℕ0 → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑤)) → 𝜃) → 𝜓)))))
127	126	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)))))
128	127	exlimiv 1938	. . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
129	75, 128	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)))))
130	129	ex 416	. . . . . . . . . . . . . . . . . . . . . . . 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))))))
131	130	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))))))
132	131	elv 3414	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 → ((𝑦 + 1) = (♯‘𝑣) → (𝑦 ∈ ℕ0 → (0 < (♯‘𝑣) → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑤)) → 𝜃) → 𝜓)))))
133	132	imp 410	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣)) → (𝑦 ∈ ℕ0 → (0 < (♯‘𝑣) → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑤)) → 𝜃) → 𝜓))))
134	133	impcom 411	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣))) → (0 < (♯‘𝑣) → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑤)) → 𝜃) → 𝜓)))
135	74, 134	mpd 15	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣))) → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑤)) → 𝜃) → 𝜓))
136	67, 135	sylan 583	. . . . . . . . . . . . . . . . . 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))
137	136	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))
138	137	alrimivv 1936	. . . . . . . . . . . . . . . 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))
139	138	ex 416	. . . . . . . . . . . . . . 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)))
140	43, 139	syl5bi 245	. . . . . . . . . . . . . 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)))
141	15, 19, 23, 27, 32, 140	uzind 12269	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝐿 ≤ 𝑛) → ∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑣)) → 𝜓))
142	4, 6, 11, 141	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (((𝐿 ≤ (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (♯‘𝑉)) → ∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑣)) → 𝜓))
143		sbcex 3704	. . . . . . . . . . . . . . 15 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → 𝑉 ∈ V)
144		sbccom 3783	. . . . . . . . . . . . . . . 16 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ↔ [𝐸 / 𝑏][𝑉 / 𝑎]𝜌)
145		sbcex 3704	. . . . . . . . . . . . . . . 16 ⊢ ([𝐸 / 𝑏][𝑉 / 𝑎]𝜌 → 𝐸 ∈ V)
146	144, 145	sylbi 220	. . . . . . . . . . . . . . 15 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → 𝐸 ∈ V)
147	143, 146	jca 515	. . . . . . . . . . . . . 14 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
148		simpl 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → 𝑣 = 𝑉)
149		simpr 488	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → 𝑒 = 𝐸)
150	149	sbceq1d 3699	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ([𝑒 / 𝑏]𝜌 ↔ [𝐸 / 𝑏]𝜌))
151	148, 150	sbceqbid 3701	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ↔ [𝑉 / 𝑎][𝐸 / 𝑏]𝜌))
152		fveq2 6717	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = 𝑉 → (♯‘𝑣) = (♯‘𝑉))
153	152	eqeq2d 2748	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝑉 → (𝑛 = (♯‘𝑣) ↔ 𝑛 = (♯‘𝑉)))
154	153	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑛 = (♯‘𝑣) ↔ 𝑛 = (♯‘𝑉)))
155	151, 154	anbi12d 634	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑣)) ↔ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑉))))
156		fi1uzind.1	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑))
157	155, 156	imbi12d 348	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑣)) → 𝜓) ↔ (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑉)) → 𝜑)))
158	157	spc2gv 3515	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑣)) → 𝜓) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑉)) → 𝜑)))
159	158	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)))
160	159	expd 419	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑛 = (♯‘𝑉) → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑣)) → 𝜓) → 𝜑))))
161	147, 160	mpcom 38	. . . . . . . . . . . . 13 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑛 = (♯‘𝑉) → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑣)) → 𝜓) → 𝜑)))
162	161	imp 410	. . . . . . . . . . . 12 ⊢ (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑉)) → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑣)) → 𝜓) → 𝜑))
163	142, 162	syl5com 31	. . . . . . . . . . 11 ⊢ (((𝐿 ≤ (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (♯‘𝑉)) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑉)) → 𝜑))
164	163	exp31 423	. . . . . . . . . 10 ⊢ (𝐿 ≤ (♯‘𝑉) → (𝑛 ∈ ℕ0 → (𝑛 = (♯‘𝑉) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑉)) → 𝜑))))
165	164	com14 96	. . . . . . . . 9 ⊢ (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑉)) → (𝑛 ∈ ℕ0 → (𝑛 = (♯‘𝑉) → (𝐿 ≤ (♯‘𝑉) → 𝜑))))
166	165	expcom 417	. . . . . . . 8 ⊢ (𝑛 = (♯‘𝑉) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑛 ∈ ℕ0 → (𝑛 = (♯‘𝑉) → (𝐿 ≤ (♯‘𝑉) → 𝜑)))))
167	166	com24 95	. . . . . . 7 ⊢ (𝑛 = (♯‘𝑉) → (𝑛 = (♯‘𝑉) → (𝑛 ∈ ℕ0 → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (♯‘𝑉) → 𝜑)))))
168	167	pm2.43i 52	. . . . . 6 ⊢ (𝑛 = (♯‘𝑉) → (𝑛 ∈ ℕ0 → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (♯‘𝑉) → 𝜑))))
169	168	imp 410	. . . . 5 ⊢ ((𝑛 = (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (♯‘𝑉) → 𝜑)))
170	169	exlimiv 1938	. . . 4 ⊢ (∃𝑛(𝑛 = (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (♯‘𝑉) → 𝜑)))
171	1, 170	sylbi 220	. . 3 ⊢ ((♯‘𝑉) ∈ ℕ0 → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (♯‘𝑉) → 𝜑)))
172		hashcl 13923	. . 3 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0)
173	171, 172	syl11 33	. 2 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑉 ∈ Fin → (𝐿 ≤ (♯‘𝑉) → 𝜑)))
174	173	3imp 1113	1 ⊢ (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089  ∀wal 1541   = wceq 1543  ∃wex 1787   ∈ wcel 2110  Vcvv 3408  [wsbc 3694   ∖ cdif 3863  {csn 4541   class class class wbr 5053  ‘cfv 6380  (class class class)co 7213  Fincfn 8626  ℝcr 10728  0cc0 10729  1c1 10730   + caddc 10732   < clt 10867   ≤ cle 10868  ℕ₀cn0 12090  ℤcz 12176  ℤ_≥cuz 12438  ♯chash 13896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-oadd 8206  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-dju 9517  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-n0 12091  df-xnn0 12163  df-z 12177  df-uz 12439  df-fz 13096  df-hash 13897

This theorem is referenced by:  brfi1uzind  14064  opfi1uzind  14067