Theorem fi1uzind 14432

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 ordered pairs of vertices and edges) with a finite number of vertices, usually with 𝐿 = 0 (see opfi1ind 14437) 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 2811	. . . 4 ⊢ ((♯‘𝑉) ∈ ℕ0 ↔ ∃𝑛(𝑛 = (♯‘𝑉) ∧ 𝑛 ∈ ℕ0))
2		fi1uzind.l	. . . . . . . . . . . . . 14 ⊢ 𝐿 ∈ ℕ0
3		nn0z 12514	. . . . . . . . . . . . . 14 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℤ)
4	2, 3	mp1i 13	. . . . . . . . . . . . 13 ⊢ (((𝐿 ≤ (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (♯‘𝑉)) → 𝐿 ∈ ℤ)
5		nn0z 12514	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℤ)
6	5	ad2antlr 728	. . . . . . . . . . . . 13 ⊢ (((𝐿 ≤ (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (♯‘𝑉)) → 𝑛 ∈ ℤ)
7		breq2 5101	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑉) = 𝑛 → (𝐿 ≤ (♯‘𝑉) ↔ 𝐿 ≤ 𝑛))
8	7	eqcoms 2743	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (♯‘𝑉) → (𝐿 ≤ (♯‘𝑉) ↔ 𝐿 ≤ 𝑛))
9	8	biimpcd 249	. . . . . . . . . . . . . . 15 ⊢ (𝐿 ≤ (♯‘𝑉) → (𝑛 = (♯‘𝑉) → 𝐿 ≤ 𝑛))
10	9	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐿 ≤ (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) → (𝑛 = (♯‘𝑉) → 𝐿 ≤ 𝑛))
11	10	imp 406	. . . . . . . . . . . . 13 ⊢ (((𝐿 ≤ (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (♯‘𝑉)) → 𝐿 ≤ 𝑛)
12		eqeq1 2739	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐿 → (𝑥 = (♯‘𝑣) ↔ 𝐿 = (♯‘𝑣)))
13	12	anbi2d 631	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐿 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (♯‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝐿 = (♯‘𝑣))))
14	13	imbi1d 341	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐿 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (♯‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝐿 = (♯‘𝑣)) → 𝜓)))
15	14	2albidv 1925	. . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝑥 = (♯‘𝑣) ↔ 𝑦 = (♯‘𝑣)))
17	16	anbi2d 631	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (♯‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑣))))
18	17	imbi1d 341	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (♯‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑣)) → 𝜓)))
19	18	2albidv 1925	. . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 + 1) → (𝑥 = (♯‘𝑣) ↔ (𝑦 + 1) = (♯‘𝑣)))
21	20	anbi2d 631	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 + 1) → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (♯‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣))))
22	21	imbi1d 341	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑦 + 1) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (♯‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣)) → 𝜓)))
23	22	2albidv 1925	. . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑛 → (𝑥 = (♯‘𝑣) ↔ 𝑛 = (♯‘𝑣)))
25	24	anbi2d 631	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑛 → (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (♯‘𝑣)) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑣))))
26	25	imbi1d 341	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑛 → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑥 = (♯‘𝑣)) → 𝜓) ↔ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑣)) → 𝜓)))
27	26	2albidv 1925	. . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . . 17 ⊢ (𝐿 = (♯‘𝑣) ↔ (♯‘𝑣) = 𝐿)
29		fi1uzind.base	. . . . . . . . . . . . . . . . 17 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (♯‘𝑣) = 𝐿) → 𝜓)
30	28, 29	sylan2b 595	. . . . . . . . . . . . . . . 16 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝐿 = (♯‘𝑣)) → 𝜓)
31	30	gen2 1798	. . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → 𝑣 = 𝑤)
34		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → 𝑒 = 𝑓)
35	34	sbceq1d 3744	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ([𝑒 / 𝑏]𝜌 ↔ [𝑓 / 𝑏]𝜌))
36	33, 35	sbceqbid 3746	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ↔ [𝑤 / 𝑎][𝑓 / 𝑏]𝜌))
37		fveq2 6833	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝑤 → (♯‘𝑣) = (♯‘𝑤))
38	37	eqeq2d 2746	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝑤 → (𝑦 = (♯‘𝑣) ↔ 𝑦 = (♯‘𝑤)))
39	38	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝑦 = (♯‘𝑣) ↔ 𝑦 = (♯‘𝑤)))
40	36, 39	anbi12d 633	. . . . . . . . . . . . . . . . 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 344	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑣)) → 𝜓) ↔ (([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑤)) → 𝜃)))
43	42	cbval2vw 2042	. . . . . . . . . . . . . . 15 ⊢ (∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑣)) → 𝜓) ↔ ∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑤)) → 𝜃))
44		nn0ge0 12428	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐿 ∈ ℕ0 → 0 ≤ 𝐿)
45		0red 11137	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ℤ → 0 ∈ ℝ)
46		nn0re 12412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℝ)
47	2, 46	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ℤ → 𝐿 ∈ ℝ)
48		zre 12494	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℝ)
49		letr 11229	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((0 ∈ ℝ ∧ 𝐿 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((0 ≤ 𝐿 ∧ 𝐿 ≤ 𝑦) → 0 ≤ 𝑦))
50	45, 47, 48, 49	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ ℤ → ((0 ≤ 𝐿 ∧ 𝐿 ≤ 𝑦) → 0 ≤ 𝑦))
51		0nn0 12418	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 0 ∈ ℕ0
52		pm3.22 459	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((0 ≤ 𝑦 ∧ 𝑦 ∈ ℤ) → (𝑦 ∈ ℤ ∧ 0 ≤ 𝑦))
53		0z 12501	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ 0 ∈ ℤ
54		eluz1 12757	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((0 ≤ 𝑦 ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ (ℤ≥‘0))
57		eluznn0 12832	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((0 ∈ ℕ0 ∧ 𝑦 ∈ (ℤ≥‘0)) → 𝑦 ∈ ℕ0)
58	51, 56, 57	sylancr 588	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((0 ≤ 𝑦 ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℕ0)
59	58	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . . . . 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 1131	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐿 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝐿 ≤ 𝑦) → 𝑦 ∈ ℕ0)
68		eqcom 2742	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 + 1) = (♯‘𝑣) ↔ (♯‘𝑣) = (𝑦 + 1))
69		nn0p1gt0 12432	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℕ0 → 0 < (𝑦 + 1))
70	69	adantr 480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 ∈ ℕ0 ∧ (♯‘𝑣) = (𝑦 + 1)) → 0 < (𝑦 + 1))
71		simpr 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 ∈ ℕ0 ∧ (♯‘𝑣) = (𝑦 + 1)) → (♯‘𝑣) = (𝑦 + 1))
72	70, 71	breqtrrd 5125	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ ℕ0 ∧ (♯‘𝑣) = (𝑦 + 1)) → 0 < (♯‘𝑣))
73	68, 72	sylan2b 595	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ ℕ0 ∧ (𝑦 + 1) = (♯‘𝑣)) → 0 < (♯‘𝑣))
74	73	adantrl 717	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣))) → 0 < (♯‘𝑣))
75		hashgt0elex 14326	. . . . . . . . . . . . . . . . . . . . . . . . . 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 3443	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ 𝑣 ∈ V
78	77	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → 𝑣 ∈ V)
79		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → 𝑛 ∈ 𝑣)
80		simpl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → 𝑦 ∈ ℕ0)
81		hashdifsnp1 14431	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ ((𝑣 ∈ V ∧ 𝑛 ∈ 𝑣 ∧ 𝑦 ∈ ℕ0) → ((♯‘𝑣) = (𝑦 + 1) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
82	68, 81	biimtrid 242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((𝑣 ∈ V ∧ 𝑛 ∈ 𝑣 ∧ 𝑦 ∈ ℕ0) → ((𝑦 + 1) = (♯‘𝑣) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
83	78, 79, 80, 82	syl3anc 1374	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → ((𝑦 + 1) = (♯‘𝑣) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
84	83	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (𝑦 + 1) = (♯‘𝑣)) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦)
85		peano2nn0 12443	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ0)
86	85	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (𝑦 + 1) = (♯‘𝑣)) → (𝑦 + 1) ∈ ℕ0)
87	86	ad2antlr 728	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 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) → [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 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 1129	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 511	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5274	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑣 ∖ {𝑛}) ∈ V
95		fi1uzind.f	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ 𝐹 ∈ V
96		simpl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → 𝑤 = (𝑣 ∖ {𝑛}))
97		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
98	97	sbceq1d 3744	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ([𝑓 / 𝑏]𝜌 ↔ [𝐹 / 𝑏]𝜌))
99	96, 98	sbceqbid 3746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ↔ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌))
100		eqcom 2742	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (𝑦 = (♯‘𝑤) ↔ (♯‘𝑤) = 𝑦)
101		fveqeq2 6842	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 ⊢ (𝑤 = (𝑣 ∖ {𝑛}) → ((♯‘𝑤) = 𝑦 ↔ (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
102	100, 101	bitrid 283	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ⊢ (𝑤 = (𝑣 ∖ {𝑛}) → (𝑦 = (♯‘𝑤) ↔ (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
103	102	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝑦 = (♯‘𝑤) ↔ (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
104	99, 103	anbi12d 633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3553	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑤)) → 𝜃) → (([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌 ∧ (♯‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒))
109	108	expdimp 452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((∀𝑤∀𝑓(([𝑤 / 𝑎][𝑓 / 𝑏]𝜌 ∧ 𝑦 = (♯‘𝑤)) → 𝜃) ∧ [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝜌) → ((♯‘(𝑣 ∖ {𝑛})) = 𝑦 → 𝜒))
110	109	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1159	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣) ∧ 𝑛 ∈ 𝑣) ↔ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣))
112	111	anbi2i 624	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ (𝑦 + 1) = (♯‘𝑣) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)
115	93, 110, 114	syl6an 685	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 434	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1932	. . . . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . . 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 3444	. . . . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . . . . . 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 581	. . . . . . . . . . . . . . . . . 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 451	. . . . . . . . . . . . . . . . 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 1930	. . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . 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	biimtrid 242	. . . . . . . . . . . . . 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 12586	. . . . . . . . . . . . 13 ⊢ ((𝐿 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝐿 ≤ 𝑛) → ∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑣)) → 𝜓))
142	4, 6, 11, 141	syl3anc 1374	. . . . . . . . . . . 12 ⊢ (((𝐿 ≤ (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) ∧ 𝑛 = (♯‘𝑉)) → ∀𝑣∀𝑒(([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑣)) → 𝜓))
143		sbcex 3749	. . . . . . . . . . . . . . 15 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → 𝑉 ∈ V)
144		sbccom 3820	. . . . . . . . . . . . . . . 16 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ↔ [𝐸 / 𝑏][𝑉 / 𝑎]𝜌)
145		sbcex 3749	. . . . . . . . . . . . . . . 16 ⊢ ([𝐸 / 𝑏][𝑉 / 𝑎]𝜌 → 𝐸 ∈ V)
146	144, 145	sylbi 217	. . . . . . . . . . . . . . 15 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → 𝐸 ∈ V)
147	143, 146	jca 511	. . . . . . . . . . . . . 14 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
148		simpl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → 𝑣 = 𝑉)
149		simpr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → 𝑒 = 𝐸)
150	149	sbceq1d 3744	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ([𝑒 / 𝑏]𝜌 ↔ [𝐸 / 𝑏]𝜌))
151	148, 150	sbceqbid 3746	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ↔ [𝑉 / 𝑎][𝐸 / 𝑏]𝜌))
152		fveq2 6833	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = 𝑉 → (♯‘𝑣) = (♯‘𝑉))
153	152	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝑉 → (𝑛 = (♯‘𝑣) ↔ 𝑛 = (♯‘𝑉)))
154	153	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑛 = (♯‘𝑣) ↔ 𝑛 = (♯‘𝑉)))
155	151, 154	anbi12d 633	. . . . . . . . . . . . . . . . . 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 344	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ((([𝑣 / 𝑎][𝑒 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑣)) → 𝜓) ↔ (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑛 = (♯‘𝑉)) → 𝜑)))
158	157	spc2gv 3553	. . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . 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 406	. . . . . . . . . . . 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 419	. . . . . . . . . 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 413	. . . . . . . 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 406	. . . . 5 ⊢ ((𝑛 = (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (♯‘𝑉) → 𝜑)))
170	169	exlimiv 1932	. . . 4 ⊢ (∃𝑛(𝑛 = (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (♯‘𝑉) → 𝜑)))
171	1, 170	sylbi 217	. . 3 ⊢ ((♯‘𝑉) ∈ ℕ0 → ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝐿 ≤ (♯‘𝑉) → 𝜑)))
172		hashcl 14281	. . 3 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0)
173	171, 172	syl11 33	. 2 ⊢ ([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 → (𝑉 ∈ Fin → (𝐿 ≤ (♯‘𝑉) → 𝜑)))
174	173	3imp 1111	1 ⊢ (([𝑉 / 𝑎][𝐸 / 𝑏]𝜌 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3439  [wsbc 3739   ∖ cdif 3897  {csn 4579   class class class wbr 5097  ‘cfv 6491  (class class class)co 7358  Fincfn 8885  ℝcr 11027  0cc0 11028  1c1 11029   + caddc 11031   < clt 11168   ≤ cle 11169  ℕ₀cn0 12403  ℤcz 12490  ℤ_≥cuz 12753  ♯chash 14255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-oadd 8401  df-er 8635  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-dju 9815  df-card 9853  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-n0 12404  df-xnn0 12477  df-z 12491  df-uz 12754  df-fz 13426  df-hash 14256

This theorem is referenced by:  brfi1uzind  14433  opfi1uzind  14436