Theorem brfi1indALT 14545

Description: Alternate proof of brfi1ind 14544, which does not use brfi1uzind 14543. (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 14391	. . 3 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0)
2		dfclel 2814	. . . 4 ⊢ ((♯‘𝑉) ∈ ℕ0 ↔ ∃𝑛(𝑛 = (♯‘𝑉) ∧ 𝑛 ∈ ℕ0))
3		eqeq2 2746	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 0 → ((♯‘𝑣) = 𝑥 ↔ (♯‘𝑣) = 0))
4	3	anbi2d 630	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = 0)))
5	4	imbi1d 341	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 0) → 𝜓)))
6	5	2albidv 1920	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 0) → 𝜓)))
7		eqeq2 2746	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((♯‘𝑣) = 𝑥 ↔ (♯‘𝑣) = 𝑦))
8	7	anbi2d 630	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦)))
9	8	imbi1d 341	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦) → 𝜓)))
10	9	2albidv 1920	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦) → 𝜓)))
11		eqeq2 2746	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 + 1) → ((♯‘𝑣) = 𝑥 ↔ (♯‘𝑣) = (𝑦 + 1)))
12	11	anbi2d 630	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 + 1) → ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1))))
13	12	imbi1d 341	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦 + 1) → (((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → 𝜓)))
14	13	2albidv 1920	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 + 1) → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → 𝜓)))
15		eqeq2 2746	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑛 → ((♯‘𝑣) = 𝑥 ↔ (♯‘𝑣) = 𝑛))
16	15	anbi2d 630	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑛 → ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛)))
17	16	imbi1d 341	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑛 → (((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) → 𝜓)))
18	17	2albidv 1920	. . . . . . . . . . 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 1792	. . . . . . . . . . 11 ⊢ ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 0) → 𝜓)
21		breq12 5152	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝑣𝐺𝑒 ↔ 𝑤𝐺𝑓))
22		fveqeq2 6915	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑤 → ((♯‘𝑣) = 𝑦 ↔ (♯‘𝑤) = 𝑦))
23	22	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ((♯‘𝑣) = 𝑦 ↔ (♯‘𝑤) = 𝑦))
24	21, 23	anbi12d 632	. . . . . . . . . . . . . 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 344	. . . . . . . . . . . . 13 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦) → 𝜓) ↔ ((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃)))
27	26	cbval2vw 2036	. . . . . . . . . . . 12 ⊢ (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦) → 𝜓) ↔ ∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃))
28		nn0p1gt0 12552	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℕ0 → 0 < (𝑦 + 1))
29	28	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℕ0 ∧ (♯‘𝑣) = (𝑦 + 1)) → 0 < (𝑦 + 1))
30		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℕ0 ∧ (♯‘𝑣) = (𝑦 + 1)) → (♯‘𝑣) = (𝑦 + 1))
31	29, 30	breqtrrd 5175	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ0 ∧ (♯‘𝑣) = (𝑦 + 1)) → 0 < (♯‘𝑣))
32	31	adantrl 716	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1))) → 0 < (♯‘𝑣))
33		hashgt0elex 14436	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑣 ∈ V ∧ 0 < (♯‘𝑣)) → ∃𝑛 𝑛 ∈ 𝑣)
34		brfi1ind.3	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑣𝐺𝑒 ∧ 𝑛 ∈ 𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)
35		vex 3481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ 𝑣 ∈ V
36		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → 𝑛 ∈ 𝑣)
37		simpl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → 𝑦 ∈ ℕ0)
38		hashdifsnp1 14541	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑣 ∈ V ∧ 𝑛 ∈ 𝑣 ∧ 𝑦 ∈ ℕ0) → ((♯‘𝑣) = (𝑦 + 1) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
39	35, 36, 37, 38	mp3an2i 1465	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → ((♯‘𝑣) = (𝑦 + 1) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
40	39	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1)) → (♯‘(𝑣 ∖ {𝑛})) = 𝑦)
41		peano2nn0 12563	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ0)
42	41	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1)) → (𝑦 + 1) ∈ ℕ0)
43	42	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → (𝑦 + 1) ∈ ℕ0)
44		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → 𝑣𝐺𝑒)
45		simplrr 778	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → (♯‘𝑣) = (𝑦 + 1))
46		simprlr 780	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (♯‘𝑣) = (𝑦 + 1))) → 𝑛 ∈ 𝑣)
47	46	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1127	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 511	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5335	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑣 ∖ {𝑛}) ∈ V
51		brfi1ind.f	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ 𝐹 ∈ V
52		breq12 5152	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝑤𝐺𝑓 ↔ (𝑣 ∖ {𝑛})𝐺𝐹))
53		fveqeq2 6915	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑤 = (𝑣 ∖ {𝑛}) → ((♯‘𝑤) = 𝑦 ↔ (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
54	53	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ((♯‘𝑤) = 𝑦 ↔ (♯‘(𝑣 ∖ {𝑛})) = 𝑦))
55	52, 54	anbi12d 632	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ↔ (((𝑣 ∖ {𝑛})𝐺𝐹 ∧ (♯‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
58	57	spc2gv 3599	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑣 ∖ {𝑛}) ∈ V ∧ 𝐹 ∈ V) → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → (((𝑣 ∖ {𝑛})𝐺𝐹 ∧ (♯‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
59	50, 51, 58	mp2an 692	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → (((𝑣 ∖ {𝑛})𝐺𝐹 ∧ (♯‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒))
60	59	expdimp 452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) → ((♯‘(𝑣 ∖ {𝑛})) = 𝑦 → 𝜒))
61	60	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 684	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 434	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . . 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 1927	. . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . 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 3482	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣𝐺𝑒 → ((♯‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (0 < (♯‘𝑣) → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
81	80	imp 406	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → (𝑦 ∈ ℕ0 → (0 < (♯‘𝑣) → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → 𝜓))))
82	81	impcom 407	. . . . . . . . . . . . . . . 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 451	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ0 ∧ ∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃)) → ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → 𝜓))
85	84	alrimivv 1925	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ0 ∧ ∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃)) → ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → 𝜓))
86	85	ex 412	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (♯‘𝑤) = 𝑦) → 𝜃) → ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → 𝜓)))
87	27, 86	biimtrid 242	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑦) → 𝜓) → ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1)) → 𝜓)))
88	6, 10, 14, 18, 20, 87	nn0ind 12710	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0 → ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) → 𝜓))
89		brfi1ind.r	. . . . . . . . . . . . 13 ⊢ Rel 𝐺
90	89	brrelex12i 5743	. . . . . . . . . . . 12 ⊢ (𝑉𝐺𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
91		breq12 5152	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑣𝐺𝑒 ↔ 𝑉𝐺𝐸))
92		fveqeq2 6915	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝑉 → ((♯‘𝑣) = 𝑛 ↔ (♯‘𝑉) = 𝑛))
93	92	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ((♯‘𝑣) = 𝑛 ↔ (♯‘𝑉) = 𝑛))
94	91, 93	anbi12d 632	. . . . . . . . . . . . . . . 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 344	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝑛) → 𝜓) ↔ ((𝑉𝐺𝐸 ∧ (♯‘𝑉) = 𝑛) → 𝜑)))
97	96	spc2gv 3599	. . . . . . . . . . . . . 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 415	. . . . . . . . . . . 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 406	. . . . . . . . . 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 413	. . . . . . . 8 ⊢ ((♯‘𝑉) = 𝑛 → (𝑉𝐺𝐸 → (𝑛 ∈ ℕ0 → 𝜑)))
104	103	com23 86	. . . . . . 7 ⊢ ((♯‘𝑉) = 𝑛 → (𝑛 ∈ ℕ0 → (𝑉𝐺𝐸 → 𝜑)))
105	104	eqcoms 2742	. . . . . 6 ⊢ (𝑛 = (♯‘𝑉) → (𝑛 ∈ ℕ0 → (𝑉𝐺𝐸 → 𝜑)))
106	105	imp 406	. . . . 5 ⊢ ((𝑛 = (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) → (𝑉𝐺𝐸 → 𝜑))
107	106	exlimiv 1927	. . . 4 ⊢ (∃𝑛(𝑛 = (♯‘𝑉) ∧ 𝑛 ∈ ℕ0) → (𝑉𝐺𝐸 → 𝜑))
108	2, 107	sylbi 217	. . 3 ⊢ ((♯‘𝑉) ∈ ℕ0 → (𝑉𝐺𝐸 → 𝜑))
109	1, 108	syl 17	. 2 ⊢ (𝑉 ∈ Fin → (𝑉𝐺𝐸 → 𝜑))
110	109	impcom 407	1 ⊢ ((𝑉𝐺𝐸 ∧ 𝑉 ∈ Fin) → 𝜑)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1534   = wceq 1536  ∃wex 1775   ∈ wcel 2105  Vcvv 3477   ∖ cdif 3959  {csn 4630   class class class wbr 5147  Rel wrel 5693  ‘cfv 6562  (class class class)co 7430  Fincfn 8983  0cc0 11152  1c1 11153   + caddc 11155   < clt 11292  ℕ₀cn0 12523  ♯chash 14365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-n0 12524  df-xnn0 12597  df-z 12611  df-uz 12876  df-fz 13544  df-hash 14366

This theorem is referenced by: (None)