Theorem brfi1indALT 14543

Description: Alternate proof of brfi1ind 14542, which does not use brfi1uzind 14541. (Contributed by Alexander van der Vekens, 7-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
brfi1ind.r	⊢ Rel 𝐺
brfi1ind.f	⊢ 𝐹 ∈ V
brfi1ind.1	⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑))
brfi1ind.2	⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃))
brfi1ind.3	⊢ ((𝑣𝐺𝑒 ∧ 𝑛 ∈ 𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)
brfi1ind.4	⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒))
brfi1ind.base	⊢ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 0) → 𝜓)
brfi1ind.step	⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)

Assertion

Ref	Expression
brfi1indALT	⊢ ((𝑉𝐺𝐸 ∧ 𝑉 ∈ Fin) → 𝜑)

Distinct variable groups:   𝑒,𝐸,𝑛,𝑣   𝑓,𝐹,𝑤   𝑒,𝐺,𝑓,𝑛,𝑣,𝑤,𝑦   𝑒,𝑉,𝑛,𝑣   𝜓,𝑓,𝑛,𝑤,𝑦   𝜃,𝑒,𝑛,𝑣   𝜒,𝑓,𝑤   𝜑,𝑒,𝑛,𝑣

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

Proof of Theorem brfi1indALT

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hashcl 14388	. . 3 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0)
2		dfclel 2839	. . . 4 ⊢ ((♯‘𝑉) ∈ ℕ0 ↔ ∃𝑛(𝑛 = (♯‘𝑉) ∧ 𝑛 ∈ ℕ0))
3		eqeq2 2775	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 0 → ((♯‘𝑣) = 𝑥 ↔ (♯‘𝑣) = 0))
4	3	anbi2d 641	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = 0)))
5	4	imbi1d 344	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 0) → 𝜓)))
6	5	2albidv 1953	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 0) → 𝜓)))
7		eqeq2 2775	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((♯‘𝑣) = 𝑥 ↔ (♯‘𝑣) = 𝑦))
8	7	anbi2d 641	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦)))
9	8	imbi1d 344	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦) → 𝜓)))
10	9	2albidv 1953	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦) → 𝜓)))
11		eqeq2 2775	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 + 1) → ((♯‘𝑣) = 𝑥 ↔ (♯‘𝑣) = (𝑦 + 1)))
12	11	anbi2d 641	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 + 1) → ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1))))
13	12	imbi1d 344	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦 + 1) → (((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → 𝜓)))
14	13	2albidv 1953	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 + 1) → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → 𝜓)))
15		eqeq2 2775	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑛 → ((♯‘𝑣) = 𝑥 ↔ (♯‘𝑣) = 𝑛))
16	15	anbi2d 641	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑛 → ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛)))
17	16	imbi1d 344	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑛 → (((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) → 𝜓)))
18	17	2albidv 1953	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑛 → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) → 𝜓)))
19		brfi1ind.base	. . . . . . . . . . . 12 ⊢ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 0) → 𝜓)
20	19	gen2 1826	. . . . . . . . . . 11 ⊢ ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 0) → 𝜓)
21		breq12 5114	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝑣𝐺𝑒 ↔ 𝑤𝐺𝑓))
22		fveqeq2 6890	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑤 → ((♯‘𝑣) = 𝑦 ↔ (♯‘𝑤) = 𝑦))
23	22	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ((♯‘𝑣) = 𝑦 ↔ (♯‘𝑤) = 𝑦))
24	21, 23	anbi12d 643	. . . . . . . . . . . . . 14 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦) ↔ (𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦)))
25		brfi1ind.2	. . . . . . . . . . . . . 14 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃))
26	24, 25	imbi12d 347	. . . . . . . . . . . . 13 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦) → 𝜓) ↔ ((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃)))
27	26	cbval2vw 2070	. . . . . . . . . . . 12 ⊢ (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦) → 𝜓) ↔ ∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃))
28		nn0p1gt0 12528	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℕ0 → 0 < (𝑦 + 1))
29	28	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℕ0 ∧ (♯‘𝑣) = (𝑦 + 1)) → 0 < (𝑦 + 1))
30		simpr 489	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℕ0 ∧ (♯‘𝑣) = (𝑦 + 1)) → (♯‘𝑣) = (𝑦 + 1))
31	29, 30	breqtrrd 5139	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ0 ∧ (♯‘𝑣) = (𝑦 + 1)) → 0 < (♯‘𝑣))
32	31	adantrl 728	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1))) → 0 < (♯‘𝑣))
33		hashgt0elex 14433	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑣 ∈ V ∧ 0 < (♯‘𝑣)) → ∃𝑛 𝑛 ∈ 𝑣)
34		brfi1ind.3	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑣𝐺𝑒 ∧ 𝑛 ∈ 𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)
35		vex 3459	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ 𝑣 ∈ V
36		simpr 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → 𝑛 ∈ 𝑣)
37		simpl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → 𝑦 ∈ ℕ0)
38		hashdifsnp1 14539	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑣 ∈ V ∧ 𝑛 ∈ 𝑣 ∧ 𝑦 ∈ ℕ0) → ((♯‘𝑣) = (𝑦 + 1) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
39	35, 36, 37, 38	mp3an2i 1495	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → ((♯‘𝑣) = (𝑦 + 1) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
40	39	imp 411	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1)) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦)
41		peano2nn0 12539	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ0)
42	41	ad2antrr 738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1)) → (𝑦 + 1) ∈ ℕ0)
43	42	ad2antlr 739	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → (𝑦 + 1) ∈ ℕ0)
44		simpr 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → 𝑣𝐺𝑒)
45		simplrr 789	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → (♯‘𝑣) = (𝑦 + 1))
46		simprlr 791	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1))) → 𝑛 ∈ 𝑣)
47	46	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → 𝑛 ∈ 𝑣)
48	44, 45, 47	3jca 1146	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣))
49	43, 48	jca 520	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → ((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)))
50	35	difexi 5301	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑣 ∖ {𝑛}) ∈ V
51		brfi1ind.f	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ 𝐹 ∈ V
52		breq12 5114	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝑤𝐺𝑓 ↔ (𝑣 ∖ {𝑛})𝐺𝐹))
53		fveqeq2 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑤 = (𝑣 ∖ {𝑛}) → ((♯‘𝑤) = 𝑦 ↔ (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
54	53	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ((♯‘𝑤) = 𝑦 ↔ (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
55	52, 54	anbi12d 643	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) ↔ ((𝑣 ∖ {𝑛})𝐺𝐹 ∧ (♯‘(𝑣 ∖ {𝑛})) = 𝑦)))
56		brfi1ind.4	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒))
57	55, 56	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ↔ (((𝑣 ∖ {𝑛})𝐺𝐹 ∧ (♯‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
58	57	spc2gv 3559	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑣 ∖ {𝑛}) ∈ V ∧ 𝐹 ∈ V) → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → (((𝑣 ∖ {𝑛})𝐺𝐹 ∧ (♯‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
59	50, 51, 58	mp2an 704	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → (((𝑣 ∖ {𝑛})𝐺𝐹 ∧ (♯‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒))
60	59	expdimp 457	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) → ((♯‘(𝑣 ∖ {𝑛})) = 𝑦 → 𝜒))
61	60	ad2antrr 738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → ((♯‘(𝑣 ∖ {𝑛})) = 𝑦 → 𝜒))
62		brfi1ind.step	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)
63	49, 61, 62	syl6an 696	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → ((♯‘(𝑣 ∖ {𝑛})) = 𝑦 → 𝜓))
64	63	exp41 439	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → ((𝑣 ∖ {𝑛})𝐺𝐹 → (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1)) → (𝑣𝐺𝑒 → ((♯‘(𝑣 ∖ {𝑛})) = 𝑦 → 𝜓)))))
65	64	com15 102	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((♯‘(𝑣 ∖ {𝑛})) = 𝑦 → ((𝑣 ∖ {𝑛})𝐺𝐹 → (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1)) → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
66	65	com23 87	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((♯‘(𝑣 ∖ {𝑛})) = 𝑦 → (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1)) → ((𝑣 ∖ {𝑛})𝐺𝐹 → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
67	40, 66	mpcom 39	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1)) → ((𝑣 ∖ {𝑛})𝐺𝐹 → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓))))
68	67	ex 417	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → ((♯‘𝑣) = (𝑦 + 1) → ((𝑣 ∖ {𝑛})𝐺𝐹 → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
69	68	com23 87	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → ((𝑣 ∖ {𝑛})𝐺𝐹 → ((♯‘𝑣) = (𝑦 + 1) → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
70	69	ex 417	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ℕ0 → (𝑛 ∈ 𝑣 → ((𝑣 ∖ {𝑛})𝐺𝐹 → ((♯‘𝑣) = (𝑦 + 1) → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓))))))
71	70	com15 102	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑣𝐺𝑒 → (𝑛 ∈ 𝑣 → ((𝑣 ∖ {𝑛})𝐺𝐹 → ((♯‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓))))))
72	71	imp 411	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑣𝐺𝑒 ∧ 𝑛 ∈ 𝑣) → ((𝑣 ∖ {𝑛})𝐺𝐹 → ((♯‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
73	34, 72	mpd 16	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑣𝐺𝑒 ∧ 𝑛 ∈ 𝑣) → ((♯‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓))))
74	73	ex 417	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣𝐺𝑒 → (𝑛 ∈ 𝑣 → ((♯‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
75	74	com4l 93	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ 𝑣 → ((♯‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
76	75	exlimiv 1960	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃𝑛 𝑛 ∈ 𝑣 → ((♯‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
77	33, 76	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 ∈ V ∧ 0 < (♯‘𝑣)) → ((♯‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
78	77	ex 417	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 ∈ V → (0 < (♯‘𝑣) → ((♯‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓))))))
79	78	com25 100	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ∈ V → (𝑣𝐺𝑒 → ((♯‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (0 < (♯‘𝑣) → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓))))))
80	79	elv 3460	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣𝐺𝑒 → ((♯‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (0 < (♯‘𝑣) → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
81	80	imp 411	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → (𝑦 ∈ ℕ0 → (0 < (♯‘𝑣) → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓))))
82	81	impcom 412	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1))) → (0 < (♯‘𝑣) → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓)))
83	32, 82	mpd 16	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1))) → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓))
84	83	impancom 456	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ0 ∧ ∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃)) → ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → 𝜓))
85	84	alrimivv 1958	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ0 ∧ ∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃)) → ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → 𝜓))
86	85	ex 417	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → 𝜓)))
87	27, 86	biimtrid 245	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦) → 𝜓) → ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → 𝜓)))
88	6, 10, 14, 18, 20, 87	nn0ind 12686	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0 → ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) → 𝜓))
89		brfi1ind.r	. . . . . . . . . . . . 13 ⊢ Rel 𝐺
90	89	brrelex12i 5716	. . . . . . . . . . . 12 ⊢ (𝑉𝐺𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
91		breq12 5114	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑣𝐺𝑒 ↔ 𝑉𝐺𝐸))
92		fveqeq2 6890	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝑉 → ((♯‘𝑣) = 𝑛 ↔ (♯‘𝑉) = 𝑛))
93	92	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ((♯‘𝑣) = 𝑛 ↔ (♯‘𝑉) = 𝑛))
94	91, 93	anbi12d 643	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) ↔ (𝑉𝐺𝐸 ∧ (♯‘𝑉) = 𝑛)))
95		brfi1ind.1	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑))
96	94, 95	imbi12d 347	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) → 𝜓) ↔ ((𝑉𝐺𝐸 ∧ (♯‘𝑉) = 𝑛) → 𝜑)))
97	96	spc2gv 3559	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) → 𝜓) → ((𝑉𝐺𝐸 ∧ (♯‘𝑉) = 𝑛) → 𝜑)))
98	97	com23 87	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑉𝐺𝐸 ∧ (♯‘𝑉) = 𝑛) → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) → 𝜓) → 𝜑)))
99	98	expd 420	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉𝐺𝐸 → ((♯‘𝑉) = 𝑛 → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) → 𝜓) → 𝜑))))
100	90, 99	mpcom 39	. . . . . . . . . . 11 ⊢ (𝑉𝐺𝐸 → ((♯‘𝑉) = 𝑛 → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) → 𝜓) → 𝜑)))
101	100	imp 411	. . . . . . . . . 10 ⊢ ((𝑉𝐺𝐸 ∧ (♯‘𝑉) = 𝑛) → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) → 𝜓) → 𝜑))
102	88, 101	syl5 35	. . . . . . . . 9 ⊢ ((𝑉𝐺𝐸 ∧ (♯‘𝑉) = 𝑛) → (𝑛 ∈ ℕ0 → 𝜑))
103	102	expcom 418	. . . . . . . 8 ⊢ ((♯‘𝑉) = 𝑛 → (𝑉𝐺𝐸 → (𝑛 ∈ ℕ0 → 𝜑)))
104	103	com23 87	. . . . . . 7 ⊢ ((♯‘𝑉) = 𝑛 → (𝑛 ∈ ℕ0 → (𝑉𝐺𝐸 → 𝜑)))
105	104	eqcoms 2771	. . . . . 6 ⊢ (𝑛 = (♯‘𝑉) → (𝑛 ∈ ℕ0 → (𝑉𝐺𝐸 → 𝜑)))
106	105	imp 411	. . . . 5 ⊢ ((𝑛 = (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) → (𝑉𝐺𝐸 → 𝜑))
107	106	exlimiv 1960	. . . 4 ⊢ (∃𝑛(𝑛 = (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) → (𝑉𝐺𝐸 → 𝜑))
108	2, 107	sylbi 220	. . 3 ⊢ ((♯‘𝑉) ∈ ℕ0 → (𝑉𝐺𝐸 → 𝜑))
109	1, 108	syl 18	. 2 ⊢ (𝑉 ∈ Fin → (𝑉𝐺𝐸 → 𝜑))
110	109	impcom 412	1 ⊢ ((𝑉𝐺𝐸 ∧ 𝑉 ∈ Fin) → 𝜑)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143  Vcvv 3455   ∖ cdif 3902  {csn 4589   class class class wbr 5109  Rel wrel 5666  ‘cfv 6536  (class class class)co 7410  Fincfn 8939  0cc0 11095  1c1 11096   + caddc 11098   < clt 11238  ℕ₀cn0 12499  ♯chash 14362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-oadd 8453  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-dju 9883  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-n0 12500  df-xnn0 12573  df-z 12587  df-uz 12858  df-fz 13531  df-hash 14363

This theorem is referenced by: (None)