Theorem brfi1indALT 13282

Description: Alternate proof of brfi1ind 13281, which does not use brfi1uzind 13280. (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 13147	. . 3 ⊢ (𝑉 ∈ Fin → (#‘𝑉) ∈ ℕ0)
2		df-clel 2618	. . . 4 ⊢ ((#‘𝑉) ∈ ℕ0 ↔ ∃𝑛(𝑛 = (#‘𝑉) ∧ 𝑛 ∈ ℕ0))
3		eqeq2 2633	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 0 → ((#‘𝑣) = 𝑥 ↔ (#‘𝑣) = 0))
4	3	anbi2d 740	. . . . . . . . . . . . 13 ⊢ (𝑥 = 0 → ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (#‘𝑣) = 0)))
5	4	imbi1d 331	. . . . . . . . . . . 12 ⊢ (𝑥 = 0 → (((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 0) → 𝜓)))
6	5	2albidv 1851	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) → 𝜓) ↔ ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 0) → 𝜓)))
7		eqeq2 2633	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((#‘𝑣) = 𝑥 ↔ (#‘𝑣) = 𝑦))
8	7	anbi2d 740	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑦)))
9	8	imbi1d 331	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑦) → 𝜓)))
10	9	2albidv 1851	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) → 𝜓) ↔ ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑦) → 𝜓)))
11		eqeq2 2633	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑦 + 1) → ((#‘𝑣) = 𝑥 ↔ (#‘𝑣) = (𝑦 + 1)))
12	11	anbi2d 740	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑦 + 1) → ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1))))
13	12	imbi1d 331	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦 + 1) → (((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1)) → 𝜓)))
14	13	2albidv 1851	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 + 1) → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) → 𝜓) ↔ ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1)) → 𝜓)))
15		eqeq2 2633	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑛 → ((#‘𝑣) = 𝑥 ↔ (#‘𝑣) = 𝑛))
16	15	anbi2d 740	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑛 → ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) ↔ (𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛)))
17	16	imbi1d 331	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑛 → (((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑥) → 𝜓) ↔ ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓)))
18	17	2albidv 1851	. . . . . . . . . . 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 1723	. . . . . . . . . . 11 ⊢ ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 0) → 𝜓)
21		breq12 4658	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝑣𝐺𝑒 ↔ 𝑤𝐺𝑓))
22		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = 𝑤 → (#‘𝑣) = (#‘𝑤))
23	22	eqeq1d 2624	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑤 → ((#‘𝑣) = 𝑦 ↔ (#‘𝑤) = 𝑦))
24	23	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ((#‘𝑣) = 𝑦 ↔ (#‘𝑤) = 𝑦))
25	21, 24	anbi12d 747	. . . . . . . . . . . . . 14 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑦) ↔ (𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦)))
26		brfi1ind.2	. . . . . . . . . . . . . 14 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃))
27	25, 26	imbi12d 334	. . . . . . . . . . . . 13 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑦) → 𝜓) ↔ ((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃)))
28	27	cbval2v 2285	. . . . . . . . . . . 12 ⊢ (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑦) → 𝜓) ↔ ∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃))
29		nn0re 11301	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ℕ0 → 𝑦 ∈ ℝ)
30		1re 10039	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℝ
31	30	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ℕ0 → 1 ∈ ℝ)
32		nn0ge0 11318	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ℕ0 → 0 ≤ 𝑦)
33		0lt1 10550	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 < 1
34	33	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ℕ0 → 0 < 1)
35	29, 31, 32, 34	addgegt0d 10601	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℕ0 → 0 < (𝑦 + 1))
36	35	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℕ0 ∧ (#‘𝑣) = (𝑦 + 1)) → 0 < (𝑦 + 1))
37		simpr 477	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℕ0 ∧ (#‘𝑣) = (𝑦 + 1)) → (#‘𝑣) = (𝑦 + 1))
38	36, 37	breqtrrd 4681	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ0 ∧ (#‘𝑣) = (𝑦 + 1)) → 0 < (#‘𝑣))
39	38	adantrl 752	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1))) → 0 < (#‘𝑣))
40		vex 3203	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑣 ∈ V
41		hashgt0elex 13189	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑣 ∈ V ∧ 0 < (#‘𝑣)) → ∃𝑛 𝑛 ∈ 𝑣)
42		brfi1ind.3	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑣𝐺𝑒 ∧ 𝑛 ∈ 𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)
43	40	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → 𝑣 ∈ V)
44		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → 𝑛 ∈ 𝑣)
45		simpl 473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → 𝑦 ∈ ℕ0)
46		brfi1indlem 13278	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑣 ∈ V ∧ 𝑛 ∈ 𝑣 ∧ 𝑦 ∈ ℕ0) → ((#‘𝑣) = (𝑦 + 1) → (#‘(𝑣 ∖ {𝑛})) = 𝑦))
47	43, 44, 45, 46	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → ((#‘𝑣) = (𝑦 + 1) → (#‘(𝑣 ∖ {𝑛})) = 𝑦))
48	47	imp 445	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (#‘𝑣) = (𝑦 + 1)) → (#‘(𝑣 ∖ {𝑛})) = 𝑦)
49		peano2nn0 11333	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ0)
50	49	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (#‘𝑣) = (𝑦 + 1)) → (𝑦 + 1) ∈ ℕ0)
51	50	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (#‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → (𝑦 + 1) ∈ ℕ0)
52		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (#‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → 𝑣𝐺𝑒)
53		simplrr 801	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (#‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → (#‘𝑣) = (𝑦 + 1))
54		simprlr 803	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (#‘𝑣) = (𝑦 + 1))) → 𝑛 ∈ 𝑣)
55	54	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ ((((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (#‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → 𝑛 ∈ 𝑣)
56	52, 53, 55	3jca 1242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (#‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣))
57	51, 56	jca 554	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
58		difexg 4808	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ (𝑣 ∈ V → (𝑣 ∖ {𝑛}) ∈ V)
59	40, 58	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (𝑣 ∖ {𝑛}) ∈ V
60		brfi1ind.f	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ 𝐹 ∈ V
61		breq12 4658	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝑤𝐺𝑓 ↔ (𝑣 ∖ {𝑛})𝐺𝐹))
62		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ⊢ (𝑤 = (𝑣 ∖ {𝑛}) → (#‘𝑤) = (#‘(𝑣 ∖ {𝑛})))
63	62	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ⊢ (𝑤 = (𝑣 ∖ {𝑛}) → ((#‘𝑤) = 𝑦 ↔ (#‘(𝑣 ∖ {𝑛})) = 𝑦))
64	63	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ((#‘𝑤) = 𝑦 ↔ (#‘(𝑣 ∖ {𝑛})) = 𝑦))
65	61, 64	anbi12d 747	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → ((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) ↔ ((𝑣 ∖ {𝑛})𝐺𝐹 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦)))
66		brfi1ind.4	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒))
67	65, 66	imbi12d 334	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ↔ (((𝑣 ∖ {𝑛})𝐺𝐹 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
68	67	spc2gv 3296	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ⊢ (((𝑣 ∖ {𝑛}) ∈ V ∧ 𝐹 ∈ V) → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → (((𝑣 ∖ {𝑛})𝐺𝐹 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒)))
69	59, 60, 68	mp2an 708	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → (((𝑣 ∖ {𝑛})𝐺𝐹 ∧ (#‘(𝑣 ∖ {𝑛})) = 𝑦) → 𝜒))
70	69	expdimp 453	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ ((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) → ((#‘(𝑣 ∖ {𝑛})) = 𝑦 → 𝜒))
71	70	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (#‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → ((#‘(𝑣 ∖ {𝑛})) = 𝑦 → 𝜒))
72		brfi1ind.step	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓)
73	57, 71, 72	syl6an 568	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((((∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) ∧ (𝑣 ∖ {𝑛})𝐺𝐹) ∧ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (#‘𝑣) = (𝑦 + 1))) ∧ 𝑣𝐺𝑒) → ((#‘(𝑣 ∖ {𝑛})) = 𝑦 → 𝜓))
74	73	exp41 638	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → ((𝑣 ∖ {𝑛})𝐺𝐹 → (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (#‘𝑣) = (𝑦 + 1)) → (𝑣𝐺𝑒 → ((#‘(𝑣 ∖ {𝑛})) = 𝑦 → 𝜓)))))
75	74	com15 101	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((#‘(𝑣 ∖ {𝑛})) = 𝑦 → ((𝑣 ∖ {𝑛})𝐺𝐹 → (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (#‘𝑣) = (𝑦 + 1)) → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
76	75	com23 86	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((#‘(𝑣 ∖ {𝑛})) = 𝑦 → (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (#‘𝑣) = (𝑦 + 1)) → ((𝑣 ∖ {𝑛})𝐺𝐹 → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
77	48, 76	mpcom 38	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) ∧ (#‘𝑣) = (𝑦 + 1)) → ((𝑣 ∖ {𝑛})𝐺𝐹 → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓))))
78	77	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → ((#‘𝑣) = (𝑦 + 1) → ((𝑣 ∖ {𝑛})𝐺𝐹 → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
79	78	com23 86	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝑛 ∈ 𝑣) → ((𝑣 ∖ {𝑛})𝐺𝐹 → ((#‘𝑣) = (𝑦 + 1) → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
80	79	ex 450	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ ℕ0 → (𝑛 ∈ 𝑣 → ((𝑣 ∖ {𝑛})𝐺𝐹 → ((#‘𝑣) = (𝑦 + 1) → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓))))))
81	80	com15 101	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑣𝐺𝑒 → (𝑛 ∈ 𝑣 → ((𝑣 ∖ {𝑛})𝐺𝐹 → ((#‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓))))))
82	81	imp 445	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑣𝐺𝑒 ∧ 𝑛 ∈ 𝑣) → ((𝑣 ∖ {𝑛})𝐺𝐹 → ((#‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
83	42, 82	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑣𝐺𝑒 ∧ 𝑛 ∈ 𝑣) → ((#‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓))))
84	83	ex 450	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣𝐺𝑒 → (𝑛 ∈ 𝑣 → ((#‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
85	84	com4l 92	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 ∈ 𝑣 → ((#‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
86	85	exlimiv 1858	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃𝑛 𝑛 ∈ 𝑣 → ((#‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
87	41, 86	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 ∈ V ∧ 0 < (#‘𝑣)) → ((#‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
88	87	ex 450	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 ∈ V → (0 < (#‘𝑣) → ((#‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (𝑣𝐺𝑒 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓))))))
89	88	com25 99	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ∈ V → (𝑣𝐺𝑒 → ((#‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (0 < (#‘𝑣) → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓))))))
90	40, 89	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣𝐺𝑒 → ((#‘𝑣) = (𝑦 + 1) → (𝑦 ∈ ℕ0 → (0 < (#‘𝑣) → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))))
91	90	imp 445	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1)) → (𝑦 ∈ ℕ0 → (0 < (#‘𝑣) → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓))))
92	91	impcom 446	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1))) → (0 < (#‘𝑣) → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓)))
93	39, 92	mpd 15	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1))) → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → 𝜓))
94	93	impancom 456	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ0 ∧ ∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃)) → ((𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1)) → 𝜓))
95	94	alrimivv 1856	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ0 ∧ ∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃)) → ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1)) → 𝜓))
96	95	ex 450	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0 → (∀𝑤∀𝑓((𝑤𝐺𝑓 ∧ (#‘𝑤) = 𝑦) → 𝜃) → ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1)) → 𝜓)))
97	28, 96	syl5bi 232	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0 → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑦) → 𝜓) → ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1)) → 𝜓)))
98	6, 10, 14, 18, 20, 97	nn0ind 11472	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0 → ∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓))
99		brfi1ind.r	. . . . . . . . . . . . . 14 ⊢ Rel 𝐺
100	99	brrelexi 5158	. . . . . . . . . . . . 13 ⊢ (𝑉𝐺𝐸 → 𝑉 ∈ V)
101	99	brrelex2i 5159	. . . . . . . . . . . . 13 ⊢ (𝑉𝐺𝐸 → 𝐸 ∈ V)
102	100, 101	jca 554	. . . . . . . . . . . 12 ⊢ (𝑉𝐺𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
103		breq12 4658	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑣𝐺𝑒 ↔ 𝑉𝐺𝐸))
104		fveq2 6191	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝑉 → (#‘𝑣) = (#‘𝑉))
105	104	eqeq1d 2624	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 = 𝑉 → ((#‘𝑣) = 𝑛 ↔ (#‘𝑉) = 𝑛))
106	105	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ((#‘𝑣) = 𝑛 ↔ (#‘𝑉) = 𝑛))
107	103, 106	anbi12d 747	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) ↔ (𝑉𝐺𝐸 ∧ (#‘𝑉) = 𝑛)))
108		brfi1ind.1	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑))
109	107, 108	imbi12d 334	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓) ↔ ((𝑉𝐺𝐸 ∧ (#‘𝑉) = 𝑛) → 𝜑)))
110	109	spc2gv 3296	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓) → ((𝑉𝐺𝐸 ∧ (#‘𝑉) = 𝑛) → 𝜑)))
111	110	com23 86	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ((𝑉𝐺𝐸 ∧ (#‘𝑉) = 𝑛) → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓) → 𝜑)))
112	111	expd 452	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉𝐺𝐸 → ((#‘𝑉) = 𝑛 → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓) → 𝜑))))
113	102, 112	mpcom 38	. . . . . . . . . . 11 ⊢ (𝑉𝐺𝐸 → ((#‘𝑉) = 𝑛 → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓) → 𝜑)))
114	113	imp 445	. . . . . . . . . 10 ⊢ ((𝑉𝐺𝐸 ∧ (#‘𝑉) = 𝑛) → (∀𝑣∀𝑒((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝑛) → 𝜓) → 𝜑))
115	98, 114	syl5 34	. . . . . . . . 9 ⊢ ((𝑉𝐺𝐸 ∧ (#‘𝑉) = 𝑛) → (𝑛 ∈ ℕ0 → 𝜑))
116	115	expcom 451	. . . . . . . 8 ⊢ ((#‘𝑉) = 𝑛 → (𝑉𝐺𝐸 → (𝑛 ∈ ℕ0 → 𝜑)))
117	116	com23 86	. . . . . . 7 ⊢ ((#‘𝑉) = 𝑛 → (𝑛 ∈ ℕ0 → (𝑉𝐺𝐸 → 𝜑)))
118	117	eqcoms 2630	. . . . . 6 ⊢ (𝑛 = (#‘𝑉) → (𝑛 ∈ ℕ0 → (𝑉𝐺𝐸 → 𝜑)))
119	118	imp 445	. . . . 5 ⊢ ((𝑛 = (#‘𝑉) ∧ 𝑛 ∈ ℕ0) → (𝑉𝐺𝐸 → 𝜑))
120	119	exlimiv 1858	. . . 4 ⊢ (∃𝑛(𝑛 = (#‘𝑉) ∧ 𝑛 ∈ ℕ0) → (𝑉𝐺𝐸 → 𝜑))
121	2, 120	sylbi 207	. . 3 ⊢ ((#‘𝑉) ∈ ℕ0 → (𝑉𝐺𝐸 → 𝜑))
122	1, 121	syl 17	. 2 ⊢ (𝑉 ∈ Fin → (𝑉𝐺𝐸 → 𝜑))
123	122	impcom 446	1 ⊢ ((𝑉𝐺𝐸 ∧ 𝑉 ∈ Fin) → 𝜑)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571  {csn 4177   class class class wbr 4653  Rel wrel 5119  ‘cfv 5888  (class class class)co 6650  Fincfn 7955  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   < clt 10074  ℕ₀cn0 11292  #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118

This theorem is referenced by: (None)