Theorem fi1uzind 13516

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 13521) 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 2813	. . . 4 ⊢ ((♯‘𝑉) ∈ ℕ0 ↔ ∃𝑛(𝑛 = (♯‘𝑉) ∧ 𝑛 ∈ ℕ0))
2		fi1uzind.l	. . . . . . . . . . . . . 14 ⊢ 𝐿 ∈ ℕ0
3		nn0z 11686	. . . . . . . . . . . . . 14 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℤ)
4	2, 3	mp1i 13	. . . . . . . . . . . . 13 ⊢ (((𝐿 ≤ (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (♯‘𝑉)) → 𝐿 ∈ ℤ)
5		nn0z 11686	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℤ)
6	5	ad2antlr 709	. . . . . . . . . . . . 13 ⊢ (((𝐿 ≤ (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (♯‘𝑉)) → 𝑛 ∈ ℤ)
7		breq2 4859	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑉) = 𝑛 → (𝐿 ≤ (♯‘𝑉) ↔ 𝐿 ≤ 𝑛))
8	7	eqcoms 2825	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (♯‘𝑉) → (𝐿 ≤ (♯‘𝑉) ↔ 𝐿 ≤ 𝑛))
9	8	biimpcd 240	. . . . . . . . . . . . . . 15 ⊢ (𝐿 ≤ (♯‘𝑉) → (𝑛 = (♯‘𝑉) → 𝐿 ≤ 𝑛))
10	9	adantr 468	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ≤ (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) → (𝑛 = (♯‘𝑉) → 𝐿 ≤ 𝑛))
11	10	imp 395	. . . . . . . . . . . . 13 ⊢ (((𝐿 ≤ (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (♯‘𝑉)) → 𝐿 ≤ 𝑛)
12		eqeq1 2821	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐿 → (𝑥 = (♯‘𝑣) ↔ 𝐿 = (♯‘𝑣)))
13	12	anbi2d 616	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐿 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (♯‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝐿 = (♯‘𝑣))))
14	13	imbi1d 332	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐿 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (♯‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝐿 = (♯‘𝑣)) → 𝜓)))
15	14	2albidv 2014	. . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝑥 = (♯‘𝑣) ↔ 𝑦 = (♯‘𝑣)))
17	16	anbi2d 616	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (♯‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑣))))
18	17	imbi1d 332	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (♯‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑣)) → 𝜓)))
19	18	2albidv 2014	. . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 = (♯‘𝑣) ↔ (𝑦 + 1) = (♯‘𝑣)))
21	20	anbi2d 616	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 + 1) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (♯‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣))))
22	21	imbi1d 332	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑦 + 1) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (♯‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣)) → 𝜓)))
23	22	2albidv 2014	. . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑛 → (𝑥 = (♯‘𝑣) ↔ 𝑛 = (♯‘𝑣)))
25	24	anbi2d 616	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑛 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (♯‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑣))))
26	25	imbi1d 332	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑛 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (♯‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑣)) → 𝜓)))
27	26	2albidv 2014	. . . . . . . . . . . . . 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 2824	. . . . . . . . . . . . . . . . 17 ⊢ (𝐿 = (♯‘𝑣) ↔ (♯‘𝑣) = 𝐿)
29		fi1uzind.base	. . . . . . . . . . . . . . . . 17 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (♯‘𝑣) = 𝐿) → 𝜓)
30	28, 29	sylan2b 583	. . . . . . . . . . . . . . . 16 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝐿 = (♯‘𝑣)) → 𝜓)
31	30	gen2 1878	. . . . . . . . . . . . . . 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 470	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → 𝑣 = 𝑤)
34		simpr 473	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → 𝑒 = 𝑓)
35	34	sbceq1d 3649	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ([𝑒 / 𝑏]𝜌 ↔ [𝑓 / 𝑏]𝜌))
36	33, 35	sbceqbid 3651	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ↔ [𝑤 / 𝑎][𝑓 / 𝑏]𝜌))
37		fveq2 6418	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝑤 → (♯‘𝑣) = (♯‘𝑤))
38	37	eqeq2d 2827	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝑤 → (𝑦 = (♯‘𝑣) ↔ 𝑦 = (♯‘𝑤)))
39	38	adantr 468	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝑦 = (♯‘𝑣) ↔ 𝑦 = (♯‘𝑤)))
40	36, 39	anbi12d 618	. . . . . . . . . . . . . . . . 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 335	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑣)) → 𝜓) ↔ (([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑤)) → 𝜃)))
43	42	cbval2v 2456	. . . . . . . . . . . . . . 15 ⊢ (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑣)) → 𝜓) ↔ ∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑤)) → 𝜃))
44		nn0ge0 11604	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐿 ∈ ℕ0 → 0 ≤ 𝐿)
45		0red 10338	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ℤ → 0 ∈ ℝ)
46		nn0re 11588	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℝ)
47	2, 46	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ℤ → 𝐿 ∈ ℝ)
48		zre 11667	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℝ)
49		letr 10426	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((0 ∈ ℝ ∧ 𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((0 ≤ 𝐿 ∧ 𝐿 ≤ 𝑦) → 0 ≤ 𝑦))
50	45, 47, 48, 49	syl3anc 1483	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ ℤ → ((0 ≤ 𝐿 ∧ 𝐿 ≤ 𝑦) → 0 ≤ 𝑦))
51		0nn0 11594	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 0 ∈ ℕ0
52		pm3.22 449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((0 ≤ 𝑦 ∧ 𝑦 ∈ ℤ) → (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦))
53		0z 11674	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 0 ∈ ℤ
54		eluz1 11928	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 248	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((0 ≤ 𝑦 ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ (ℤ≥‘0))
57		eluznn0 11996	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((0 ∈ ℕ0 ∧ 𝑦 ∈ (ℤ≥‘0)) → 𝑦 ∈ ℕ0)
58	51, 56, 57	sylancr 577	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((0 ≤ 𝑦 ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℕ0)
59	58	ex 399	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 399	. . . . . . . . . . . . . . . . . . . . . . . . 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 395	. . . . . . . . . . . . . . . . . . . . . 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 1153	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿 ≤ 𝑦) → 𝑦 ∈ ℕ0)
68		eqcom 2824	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 + 1) = (♯‘𝑣) ↔ (♯‘𝑣) = (𝑦 + 1))
69		nn0p1gt0 11608	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℕ0 → 0 < (𝑦 + 1))
70	69	adantr 468	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 ∈ ℕ0 ∧ (♯‘𝑣) = (𝑦 + 1)) → 0 < (𝑦 + 1))
71		simpr 473	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 ∈ ℕ0 ∧ (♯‘𝑣) = (𝑦 + 1)) → (♯‘𝑣) = (𝑦 + 1))
72	70, 71	breqtrrd 4883	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ ℕ0 ∧ (♯‘𝑣) = (𝑦 + 1)) → 0 < (♯‘𝑣))
73	68, 72	sylan2b 583	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ ℕ0 ∧ (𝑦 + 1) = (♯‘𝑣)) → 0 < (♯‘𝑣))
74	73	adantrl 698	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣))) → 0 < (♯‘𝑣))
75		vex 3405	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑣 ∈ V
76		hashgt0elex 13426	. . . . . . . . . . . . . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → 𝑛 ∈ 𝑣)
80		simpl 470	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → 𝑦 ∈ ℕ0)
81		hashdifsnp1 13515	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑣 ∈ V ∧ 𝑛 ∈ 𝑣 ∧ 𝑦 ∈ ℕ0) → ((♯‘𝑣) = (𝑦 + 1) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
82	68, 81	syl5bi 233	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑣 ∈ V ∧ 𝑛 ∈ 𝑣 ∧ 𝑦 ∈ ℕ0) → ((𝑦 + 1) = (♯‘𝑣) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
83	78, 79, 80, 82	syl3anc 1483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → ((𝑦 + 1) = (♯‘𝑣) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
84	83	imp 395	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (𝑦 + 1) = (♯‘𝑣)) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦)
85		peano2nn0 11619	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ0)
86	85	ad2antrr 708	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (𝑦 + 1) = (♯‘𝑣)) → (𝑦 + 1) ∈ ℕ0)
87	86	ad2antlr 709	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 468	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1151	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 503	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5016	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑣 ∈ V → (𝑣 ∖ {𝑛}) ∈ V)
95	75, 94	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑣 ∖ {𝑛}) ∈ V
96		fi1uzind.f	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ 𝐹 ∈ V
97		simpl 470	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → 𝑤 = (𝑣 ∖ {𝑛}))
98		simpr 473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
99	98	sbceq1d 3649	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ([𝑓 / 𝑏]𝜌 ↔ [𝐹 / 𝑏]𝜌))
100	97, 99	sbceqbid 3651	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ↔ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌))
101		eqcom 2824	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (𝑦 = (♯‘𝑤) ↔ (♯‘𝑤) = 𝑦)
102		fveqeq2 6427	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (𝑤 = (𝑣 ∖ {𝑛}) → ((♯‘𝑤) = 𝑦 ↔ (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
103	101, 102	syl5bb 274	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (𝑤 = (𝑣 ∖ {𝑛}) → (𝑦 = (♯‘𝑤) ↔ (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
104	103	adantr 468	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝑦 = (♯‘𝑤) ↔ (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
105	100, 104	anbi12d 618	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑤)) ↔ ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (♯‘(𝑣 ∖ {𝑛})) = 𝑦)))
106		fi1uzind.4	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒))
107	105, 106	imbi12d 335	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ((([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑤)) → 𝜃) ↔ (([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (♯‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
108	107	spc2gv 3500	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
109	95, 96, 108	mp2an 675	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑤)) → 𝜃) → (([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (♯‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒))
110	109	expdimp 442	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) → ((♯‘(𝑣 ∖ {𝑛})) = 𝑦 → 𝜒))
111	110	ad2antrr 708	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
112	68	3anbi2i 1190	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣) ∧ 𝑛 ∈ 𝑣) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣))
113	112	anbi2i 611	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
114		fi1uzind.step	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)
115	113, 114	sylanb 572	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)
116	93, 111, 115	syl6an 666	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
117	116	exp41 423	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
118	117	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)))))
119	118	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)))))
120	84, 119	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))))
121	120	ex 399	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
122	121	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)))))
123	122	ex 399	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))))
124	123	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))))))
125	124	imp 395	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
126	77, 125	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))))
127	126	ex 399	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ([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	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)))))
129	128	exlimiv 2021	. . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
130	76, 129	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)))))
131	130	ex 399	. . . . . . . . . . . . . . . . . . . . . . . 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))))))
132	131	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))))))
133	75, 132	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)))))
134	133	imp 395	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (([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	impcom 396	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣))) → (0 < (♯‘𝑣) → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑤)) → 𝜃) → 𝜓)))
136	74, 135	mpd 15	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣))) → (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑤)) → 𝜃) → 𝜓))
137	67, 136	sylan 571	. . . . . . . . . . . . . . . . . 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))
138	137	impancom 441	. . . . . . . . . . . . . . . . 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))
139	138	alrimivv 2019	. . . . . . . . . . . . . . . 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))
140	139	ex 399	. . . . . . . . . . . . . . 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)))
141	43, 140	syl5bi 233	. . . . . . . . . . . . . 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)))
142	15, 19, 23, 27, 32, 141	uzind 11755	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝐿 ≤ 𝑛) → ∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑣)) → 𝜓))
143	4, 6, 11, 142	syl3anc 1483	. . . . . . . . . . . 12 ⊢ (((𝐿 ≤ (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (♯‘𝑉)) → ∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑣)) → 𝜓))
144		sbcex 3654	. . . . . . . . . . . . . . 15 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → 𝑉 ∈ V)
145		sbccom 3716	. . . . . . . . . . . . . . . 16 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ↔ [𝐸 / 𝑏][𝑉 / 𝑎]𝜌)
146		sbcex 3654	. . . . . . . . . . . . . . . 16 ⊢ ([𝐸 / 𝑏][𝑉 / 𝑎]𝜌 → 𝐸 ∈ V)
147	145, 146	sylbi 208	. . . . . . . . . . . . . . 15 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → 𝐸 ∈ V)
148	144, 147	jca 503	. . . . . . . . . . . . . 14 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
149		simpl 470	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → 𝑣 = 𝑉)
150		simpr 473	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → 𝑒 = 𝐸)
151	150	sbceq1d 3649	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ([𝑒 / 𝑏]𝜌 ↔ [𝐸 / 𝑏]𝜌))
152	149, 151	sbceqbid 3651	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ↔ [𝑉 / 𝑎][𝐸 / 𝑏]𝜌))
153		fveq2 6418	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = 𝑉 → (♯‘𝑣) = (♯‘𝑉))
154	153	eqeq2d 2827	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝑉 → (𝑛 = (♯‘𝑣) ↔ 𝑛 = (♯‘𝑉)))
155	154	adantr 468	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑛 = (♯‘𝑣) ↔ 𝑛 = (♯‘𝑉)))
156	152, 155	anbi12d 618	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑣)) ↔ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑉))))
157		fi1uzind.1	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑))
158	156, 157	imbi12d 335	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑣)) → 𝜓) ↔ (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑉)) → 𝜑)))
159	158	spc2gv 3500	. . . . . . . . . . . . . . . 16 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑣)) → 𝜓) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑉)) → 𝜑)))
160	159	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)))
161	160	expd 402	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑛 = (♯‘𝑉) → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑣)) → 𝜓) → 𝜑))))
162	148, 161	mpcom 38	. . . . . . . . . . . . 13 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑛 = (♯‘𝑉) → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑣)) → 𝜓) → 𝜑)))
163	162	imp 395	. . . . . . . . . . . 12 ⊢ (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑉)) → (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑣)) → 𝜓) → 𝜑))
164	143, 163	syl5com 31	. . . . . . . . . . 11 ⊢ (((𝐿 ≤ (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (♯‘𝑉)) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑉)) → 𝜑))
165	164	exp31 408	. . . . . . . . . 10 ⊢ (𝐿 ≤ (♯‘𝑉) → (𝑛 ∈ ℕ0 → (𝑛 = (♯‘𝑉) → (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑉)) → 𝜑))))
166	165	com14 96	. . . . . . . . 9 ⊢ (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑉)) → (𝑛 ∈ ℕ0 → (𝑛 = (♯‘𝑉) → (𝐿 ≤ (♯‘𝑉) → 𝜑))))
167	166	expcom 400	. . . . . . . 8 ⊢ (𝑛 = (♯‘𝑉) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑛 ∈ ℕ0 → (𝑛 = (♯‘𝑉) → (𝐿 ≤ (♯‘𝑉) → 𝜑)))))
168	167	com24 95	. . . . . . 7 ⊢ (𝑛 = (♯‘𝑉) → (𝑛 = (♯‘𝑉) → (𝑛 ∈ ℕ0 → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (♯‘𝑉) → 𝜑)))))
169	168	pm2.43i 52	. . . . . 6 ⊢ (𝑛 = (♯‘𝑉) → (𝑛 ∈ ℕ0 → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (♯‘𝑉) → 𝜑))))
170	169	imp 395	. . . . 5 ⊢ ((𝑛 = (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (♯‘𝑉) → 𝜑)))
171	170	exlimiv 2021	. . . 4 ⊢ (∃𝑛(𝑛 = (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (♯‘𝑉) → 𝜑)))
172	1, 171	sylbi 208	. . 3 ⊢ ((♯‘𝑉) ∈ ℕ0 → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (♯‘𝑉) → 𝜑)))
173		hashcl 13385	. . 3 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0)
174	172, 173	syl11 33	. 2 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑉 ∈ Fin → (𝐿 ≤ (♯‘𝑉) → 𝜑)))
175	174	3imp 1130	1 ⊢ (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑)

Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384   ∧ w3a 1100  ∀wal 1635   = wceq 1637  ∃wex 1859   ∈ wcel 2157  Vcvv 3402  [wsbc 3644   ∖ cdif 3777  {csn 4381   class class class wbr 4855  ‘cfv 6111  (class class class)co 6884  Fincfn 8202  ℝcr 10230  0cc0 10231  1c1 10232   + caddc 10234   < clt 10369   ≤ cle 10370  ℕ₀cn0 11579  ℤcz 11663  ℤ_≥cuz 11924  ♯chash 13357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7189  ax-cnex 10287  ax-resscn 10288  ax-1cn 10289  ax-icn 10290  ax-addcl 10291  ax-addrcl 10292  ax-mulcl 10293  ax-mulrcl 10294  ax-mulcom 10295  ax-addass 10296  ax-mulass 10297  ax-distr 10298  ax-i2m1 10299  ax-1ne0 10300  ax-1rid 10301  ax-rnegex 10302  ax-rrecex 10303  ax-cnre 10304  ax-pre-lttri 10305  ax-pre-lttrn 10306  ax-pre-ltadd 10307  ax-pre-mulgt0 10308

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5232  df-eprel 5237  df-po 5245  df-so 5246  df-fr 5283  df-we 5285  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-pred 5907  df-ord 5953  df-on 5954  df-lim 5955  df-suc 5956  df-iota 6074  df-fun 6113  df-fn 6114  df-f 6115  df-f1 6116  df-fo 6117  df-f1o 6118  df-fv 6119  df-riota 6845  df-ov 6887  df-oprab 6888  df-mpt2 6889  df-om 7306  df-1st 7408  df-2nd 7409  df-wrecs 7652  df-recs 7714  df-rdg 7752  df-1o 7806  df-oadd 7810  df-er 7989  df-en 8203  df-dom 8204  df-sdom 8205  df-fin 8206  df-card 9058  df-cda 9285  df-pnf 10371  df-mnf 10372  df-xr 10373  df-ltxr 10374  df-le 10375  df-sub 10563  df-neg 10564  df-nn 11316  df-n0 11580  df-xnn0 11650  df-z 11664  df-uz 11925  df-fz 12570  df-hash 13358

This theorem is referenced by:  brfi1uzind  13517  opfi1uzind  13520