Theorem brfi1indALT 14491

Description: Alternate proof of brfi1ind 14490, which does not use brfi1uzind 14489. (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 14345	. . 3 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0)
2		dfclel 2803	. . . 4 ⊢ ((♯‘𝑉) ∈ ℕ0 ↔ ∃𝑛(𝑛 = (♯‘𝑉) ∧ 𝑛 ∈ ℕ0))
3		eqeq2 2737	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 0 → ((♯‘𝑣) = 𝑥 ↔ (♯‘𝑣) = 0))
4	3	anbi2d 628	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = 0)))
5	4	imbi1d 340	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 0) → 𝜓)))
6	5	2albidv 1918	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 0) → 𝜓)))
7		eqeq2 2737	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((♯‘𝑣) = 𝑥 ↔ (♯‘𝑣) = 𝑦))
8	7	anbi2d 628	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦)))
9	8	imbi1d 340	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦) → 𝜓)))
10	9	2albidv 1918	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦) → 𝜓)))
11		eqeq2 2737	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 + 1) → ((♯‘𝑣) = 𝑥 ↔ (♯‘𝑣) = (𝑦 + 1)))
12	11	anbi2d 628	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 + 1) → ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1))))
13	12	imbi1d 340	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦 + 1) → (((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → 𝜓)))
14	13	2albidv 1918	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 + 1) → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → 𝜓)))
15		eqeq2 2737	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑛 → ((♯‘𝑣) = 𝑥 ↔ (♯‘𝑣) = 𝑛))
16	15	anbi2d 628	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑛 → ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛)))
17	16	imbi1d 340	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑛 → (((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) → 𝜓)))
18	17	2albidv 1918	. . . . . . . . . . 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 1790	. . . . . . . . . . 11 ⊢ ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 0) → 𝜓)
21		breq12 5148	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝑣𝐺𝑒 ↔ 𝑤𝐺𝑓))
22		fveqeq2 6900	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑤 → ((♯‘𝑣) = 𝑦 ↔ (♯‘𝑤) = 𝑦))
23	22	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ((♯‘𝑣) = 𝑦 ↔ (♯‘𝑤) = 𝑦))
24	21, 23	anbi12d 630	. . . . . . . . . . . . . 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 343	. . . . . . . . . . . . 13 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦) → 𝜓) ↔ ((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃)))
27	26	cbval2vw 2035	. . . . . . . . . . . 12 ⊢ (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦) → 𝜓) ↔ ∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃))
28		nn0p1gt0 12529	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℕ0 → 0 < (𝑦 + 1))
29	28	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℕ0 ∧ (♯‘𝑣) = (𝑦 + 1)) → 0 < (𝑦 + 1))
30		simpr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℕ0 ∧ (♯‘𝑣) = (𝑦 + 1)) → (♯‘𝑣) = (𝑦 + 1))
31	29, 30	breqtrrd 5171	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ0 ∧ (♯‘𝑣) = (𝑦 + 1)) → 0 < (♯‘𝑣))
32	31	adantrl 714	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1))) → 0 < (♯‘𝑣))
33		hashgt0elex 14390	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑣 ∈ V ∧ 0 < (♯‘𝑣)) → ∃𝑛 𝑛 ∈ 𝑣)
34		brfi1ind.3	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑣𝐺𝑒 ∧ 𝑛 ∈ 𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)
35		vex 3467	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ 𝑣 ∈ V
36		simpr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → 𝑛 ∈ 𝑣)
37		simpl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → 𝑦 ∈ ℕ0)
38		hashdifsnp1 14487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑣 ∈ V ∧ 𝑛 ∈ 𝑣 ∧ 𝑦 ∈ ℕ0) → ((♯‘𝑣) = (𝑦 + 1) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
39	35, 36, 37, 38	mp3an2i 1462	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → ((♯‘𝑣) = (𝑦 + 1) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
40	39	imp 405	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1)) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦)
41		peano2nn0 12540	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ0)
42	41	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1)) → (𝑦 + 1) ∈ ℕ0)
43	42	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → (𝑦 + 1) ∈ ℕ0)
44		simpr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → 𝑣𝐺𝑒)
45		simplrr 776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → (♯‘𝑣) = (𝑦 + 1))
46		simprlr 778	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1))) → 𝑛 ∈ 𝑣)
47	46	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1125	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 510	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5325	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑣 ∖ {𝑛}) ∈ V
51		brfi1ind.f	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ 𝐹 ∈ V
52		breq12 5148	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝑤𝐺𝑓 ↔ (𝑣 ∖ {𝑛})𝐺𝐹))
53		fveqeq2 6900	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑤 = (𝑣 ∖ {𝑛}) → ((♯‘𝑤) = 𝑦 ↔ (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
54	53	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ((♯‘𝑤) = 𝑦 ↔ (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
55	52, 54	anbi12d 630	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 343	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ↔ (((𝑣 ∖ {𝑛})𝐺𝐹 ∧ (♯‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
58	57	spc2gv 3580	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑣 ∖ {𝑛}) ∈ V ∧ 𝐹 ∈ V) → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → (((𝑣 ∖ {𝑛})𝐺𝐹 ∧ (♯‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
59	50, 51, 58	mp2an 690	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → (((𝑣 ∖ {𝑛})𝐺𝐹 ∧ (♯‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒))
60	59	expdimp 451	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) → ((♯‘(𝑣 ∖ {𝑛})) = 𝑦 → 𝜒))
61	60	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 682	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 433	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 101	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 86	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 38	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1)) → ((𝑣 ∖ {𝑛})𝐺𝐹 → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓))))
68	67	ex 411	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → ((♯‘𝑣) = (𝑦 + 1) → ((𝑣 ∖ {𝑛})𝐺𝐹 → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
69	68	com23 86	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → ((𝑣 ∖ {𝑛})𝐺𝐹 → ((♯‘𝑣) = (𝑦 + 1) → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
70	69	ex 411	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ℕ0 → (𝑛 ∈ 𝑣 → ((𝑣 ∖ {𝑛})𝐺𝐹 → ((♯‘𝑣) = (𝑦 + 1) → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓))))))
71	70	com15 101	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑣𝐺𝑒 → (𝑛 ∈ 𝑣 → ((𝑣 ∖ {𝑛})𝐺𝐹 → ((♯‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓))))))
72	71	imp 405	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑣𝐺𝑒 ∧ 𝑛 ∈ 𝑣) → ((𝑣 ∖ {𝑛})𝐺𝐹 → ((♯‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
73	34, 72	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑣𝐺𝑒 ∧ 𝑛 ∈ 𝑣) → ((♯‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓))))
74	73	ex 411	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣𝐺𝑒 → (𝑛 ∈ 𝑣 → ((♯‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
75	74	com4l 92	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ 𝑣 → ((♯‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
76	75	exlimiv 1925	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃𝑛 𝑛 ∈ 𝑣 → ((♯‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
77	33, 76	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 ∈ V ∧ 0 < (♯‘𝑣)) → ((♯‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
78	77	ex 411	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 ∈ V → (0 < (♯‘𝑣) → ((♯‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓))))))
79	78	com25 99	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ∈ V → (𝑣𝐺𝑒 → ((♯‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (0 < (♯‘𝑣) → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓))))))
80	79	elv 3469	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣𝐺𝑒 → ((♯‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (0 < (♯‘𝑣) → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
81	80	imp 405	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → (𝑦 ∈ ℕ0 → (0 < (♯‘𝑣) → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓))))
82	81	impcom 406	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1))) → (0 < (♯‘𝑣) → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓)))
83	32, 82	mpd 15	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1))) → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓))
84	83	impancom 450	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ0 ∧ ∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃)) → ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → 𝜓))
85	84	alrimivv 1923	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ0 ∧ ∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃)) → ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → 𝜓))
86	85	ex 411	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → 𝜓)))
87	27, 86	biimtrid 241	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦) → 𝜓) → ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → 𝜓)))
88	6, 10, 14, 18, 20, 87	nn0ind 12685	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0 → ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) → 𝜓))
89		brfi1ind.r	. . . . . . . . . . . . 13 ⊢ Rel 𝐺
90	89	brrelex12i 5727	. . . . . . . . . . . 12 ⊢ (𝑉𝐺𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
91		breq12 5148	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑣𝐺𝑒 ↔ 𝑉𝐺𝐸))
92		fveqeq2 6900	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝑉 → ((♯‘𝑣) = 𝑛 ↔ (♯‘𝑉) = 𝑛))
93	92	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ((♯‘𝑣) = 𝑛 ↔ (♯‘𝑉) = 𝑛))
94	91, 93	anbi12d 630	. . . . . . . . . . . . . . . 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 343	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) → 𝜓) ↔ ((𝑉𝐺𝐸 ∧ (♯‘𝑉) = 𝑛) → 𝜑)))
97	96	spc2gv 3580	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) → 𝜓) → ((𝑉𝐺𝐸 ∧ (♯‘𝑉) = 𝑛) → 𝜑)))
98	97	com23 86	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑉𝐺𝐸 ∧ (♯‘𝑉) = 𝑛) → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) → 𝜓) → 𝜑)))
99	98	expd 414	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉𝐺𝐸 → ((♯‘𝑉) = 𝑛 → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) → 𝜓) → 𝜑))))
100	90, 99	mpcom 38	. . . . . . . . . . 11 ⊢ (𝑉𝐺𝐸 → ((♯‘𝑉) = 𝑛 → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) → 𝜓) → 𝜑)))
101	100	imp 405	. . . . . . . . . 10 ⊢ ((𝑉𝐺𝐸 ∧ (♯‘𝑉) = 𝑛) → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) → 𝜓) → 𝜑))
102	88, 101	syl5 34	. . . . . . . . 9 ⊢ ((𝑉𝐺𝐸 ∧ (♯‘𝑉) = 𝑛) → (𝑛 ∈ ℕ0 → 𝜑))
103	102	expcom 412	. . . . . . . 8 ⊢ ((♯‘𝑉) = 𝑛 → (𝑉𝐺𝐸 → (𝑛 ∈ ℕ0 → 𝜑)))
104	103	com23 86	. . . . . . 7 ⊢ ((♯‘𝑉) = 𝑛 → (𝑛 ∈ ℕ0 → (𝑉𝐺𝐸 → 𝜑)))
105	104	eqcoms 2733	. . . . . 6 ⊢ (𝑛 = (♯‘𝑉) → (𝑛 ∈ ℕ0 → (𝑉𝐺𝐸 → 𝜑)))
106	105	imp 405	. . . . 5 ⊢ ((𝑛 = (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) → (𝑉𝐺𝐸 → 𝜑))
107	106	exlimiv 1925	. . . 4 ⊢ (∃𝑛(𝑛 = (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) → (𝑉𝐺𝐸 → 𝜑))
108	2, 107	sylbi 216	. . 3 ⊢ ((♯‘𝑉) ∈ ℕ0 → (𝑉𝐺𝐸 → 𝜑))
109	1, 108	syl 17	. 2 ⊢ (𝑉 ∈ Fin → (𝑉𝐺𝐸 → 𝜑))
110	109	impcom 406	1 ⊢ ((𝑉𝐺𝐸 ∧ 𝑉 ∈ Fin) → 𝜑)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084  ∀wal 1531   = wceq 1533  ∃wex 1773   ∈ wcel 2098  Vcvv 3463   ∖ cdif 3937  {csn 4624   class class class wbr 5143  Rel wrel 5677  ‘cfv 6542  (class class class)co 7415  Fincfn 8960  0cc0 11136  1c1 11137   + caddc 11139   < clt 11276  ℕ₀cn0 12500  ♯chash 14319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-oadd 8487  df-er 8721  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-dju 9922  df-card 9960  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-nn 12241  df-n0 12501  df-xnn0 12573  df-z 12587  df-uz 12851  df-fz 13515  df-hash 14320

This theorem is referenced by: (None)