Theorem ac6sfi 9207

Description: A version of ac6s 10413 for finite sets. (Contributed by Jeff Hankins, 26-Jun-2009.) (Proof shortened by Mario Carneiro, 29-Jan-2014.)

Hypothesis

Ref	Expression
ac6sfi.1	⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
ac6sfi	⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))

Distinct variable groups:   𝑥,𝑓,𝐴   𝑦,𝑓,𝐵,𝑥   𝜑,𝑓   𝜓,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑓)   𝐴(𝑦)

Proof of Theorem ac6sfi

Dummy variables 𝑢 𝑤 𝑧 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		raleq 3293	. . . 4 ⊢ (𝑢 = ∅ → (∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ ∅ ∃𝑦 ∈ 𝐵 𝜑))
2		feq2 6649	. . . . . 6 ⊢ (𝑢 = ∅ → (𝑓:𝑢⟶𝐵 ↔ 𝑓:∅⟶𝐵))
3		raleq 3293	. . . . . 6 ⊢ (𝑢 = ∅ → (∀𝑥 ∈ 𝑢 𝜓 ↔ ∀𝑥 ∈ ∅ 𝜓))
4	2, 3	anbi12d 632	. . . . 5 ⊢ (𝑢 = ∅ → ((𝑓:𝑢⟶𝐵 ∧ ∀𝑥 ∈ 𝑢 𝜓) ↔ (𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓)))
5	4	exbidv 1921	. . . 4 ⊢ (𝑢 = ∅ → (∃𝑓(𝑓:𝑢⟶𝐵 ∧ ∀𝑥 ∈ 𝑢 𝜓) ↔ ∃𝑓(𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓)))
6	1, 5	imbi12d 344	. . 3 ⊢ (𝑢 = ∅ → ((∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝑢⟶𝐵 ∧ ∀𝑥 ∈ 𝑢 𝜓)) ↔ (∀𝑥 ∈ ∅ ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓))))
7		raleq 3293	. . . 4 ⊢ (𝑢 = 𝑤 → (∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝑤 ∃𝑦 ∈ 𝐵 𝜑))
8		feq2 6649	. . . . . 6 ⊢ (𝑢 = 𝑤 → (𝑓:𝑢⟶𝐵 ↔ 𝑓:𝑤⟶𝐵))
9		raleq 3293	. . . . . 6 ⊢ (𝑢 = 𝑤 → (∀𝑥 ∈ 𝑢 𝜓 ↔ ∀𝑥 ∈ 𝑤 𝜓))
10	8, 9	anbi12d 632	. . . . 5 ⊢ (𝑢 = 𝑤 → ((𝑓:𝑢⟶𝐵 ∧ ∀𝑥 ∈ 𝑢 𝜓) ↔ (𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)))
11	10	exbidv 1921	. . . 4 ⊢ (𝑢 = 𝑤 → (∃𝑓(𝑓:𝑢⟶𝐵 ∧ ∀𝑥 ∈ 𝑢 𝜓) ↔ ∃𝑓(𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)))
12	7, 11	imbi12d 344	. . 3 ⊢ (𝑢 = 𝑤 → ((∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝑢⟶𝐵 ∧ ∀𝑥 ∈ 𝑢 𝜓)) ↔ (∀𝑥 ∈ 𝑤 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓))))
13		raleq 3293	. . . 4 ⊢ (𝑢 = (𝑤 ∪ {𝑧}) → (∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦 ∈ 𝐵 𝜑))
14		feq2 6649	. . . . . . 7 ⊢ (𝑢 = (𝑤 ∪ {𝑧}) → (𝑓:𝑢⟶𝐵 ↔ 𝑓:(𝑤 ∪ {𝑧})⟶𝐵))
15		raleq 3293	. . . . . . 7 ⊢ (𝑢 = (𝑤 ∪ {𝑧}) → (∀𝑥 ∈ 𝑢 𝜓 ↔ ∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓))
16	14, 15	anbi12d 632	. . . . . 6 ⊢ (𝑢 = (𝑤 ∪ {𝑧}) → ((𝑓:𝑢⟶𝐵 ∧ ∀𝑥 ∈ 𝑢 𝜓) ↔ (𝑓:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓)))
17	16	exbidv 1921	. . . . 5 ⊢ (𝑢 = (𝑤 ∪ {𝑧}) → (∃𝑓(𝑓:𝑢⟶𝐵 ∧ ∀𝑥 ∈ 𝑢 𝜓) ↔ ∃𝑓(𝑓:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓)))
18		feq1 6648	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑓:(𝑤 ∪ {𝑧})⟶𝐵 ↔ 𝑔:(𝑤 ∪ {𝑧})⟶𝐵))
19		fvex 6853	. . . . . . . . . 10 ⊢ (𝑓‘𝑥) ∈ V
20		ac6sfi.1	. . . . . . . . . 10 ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))
21	19, 20	sbcie 3792	. . . . . . . . 9 ⊢ ([(𝑓‘𝑥) / 𝑦]𝜑 ↔ 𝜓)
22		fveq1 6839	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑥) = (𝑔‘𝑥))
23	22	sbceq1d 3755	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ([(𝑓‘𝑥) / 𝑦]𝜑 ↔ [(𝑔‘𝑥) / 𝑦]𝜑))
24	21, 23	bitr3id 285	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝜓 ↔ [(𝑔‘𝑥) / 𝑦]𝜑))
25	24	ralbidv 3156	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓 ↔ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑))
26	18, 25	anbi12d 632	. . . . . 6 ⊢ (𝑓 = 𝑔 → ((𝑓:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓) ↔ (𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑)))
27	26	cbvexvw 2037	. . . . 5 ⊢ (∃𝑓(𝑓:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓) ↔ ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑))
28	17, 27	bitrdi 287	. . . 4 ⊢ (𝑢 = (𝑤 ∪ {𝑧}) → (∃𝑓(𝑓:𝑢⟶𝐵 ∧ ∀𝑥 ∈ 𝑢 𝜓) ↔ ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑)))
29	13, 28	imbi12d 344	. . 3 ⊢ (𝑢 = (𝑤 ∪ {𝑧}) → ((∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝑢⟶𝐵 ∧ ∀𝑥 ∈ 𝑢 𝜓)) ↔ (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦 ∈ 𝐵 𝜑 → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑))))
30		raleq 3293	. . . 4 ⊢ (𝑢 = 𝐴 → (∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑))
31		feq2 6649	. . . . . 6 ⊢ (𝑢 = 𝐴 → (𝑓:𝑢⟶𝐵 ↔ 𝑓:𝐴⟶𝐵))
32		raleq 3293	. . . . . 6 ⊢ (𝑢 = 𝐴 → (∀𝑥 ∈ 𝑢 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜓))
33	31, 32	anbi12d 632	. . . . 5 ⊢ (𝑢 = 𝐴 → ((𝑓:𝑢⟶𝐵 ∧ ∀𝑥 ∈ 𝑢 𝜓) ↔ (𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)))
34	33	exbidv 1921	. . . 4 ⊢ (𝑢 = 𝐴 → (∃𝑓(𝑓:𝑢⟶𝐵 ∧ ∀𝑥 ∈ 𝑢 𝜓) ↔ ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)))
35	30, 34	imbi12d 344	. . 3 ⊢ (𝑢 = 𝐴 → ((∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝑢⟶𝐵 ∧ ∀𝑥 ∈ 𝑢 𝜓)) ↔ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))))
36		f0 6723	. . . 4 ⊢ ∅:∅⟶𝐵
37		0ex 5257	. . . . 5 ⊢ ∅ ∈ V
38		ral0 4472	. . . . . . 7 ⊢ ∀𝑥 ∈ ∅ 𝜓
39	38	biantru 529	. . . . . 6 ⊢ (𝑓:∅⟶𝐵 ↔ (𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓))
40		feq1 6648	. . . . . 6 ⊢ (𝑓 = ∅ → (𝑓:∅⟶𝐵 ↔ ∅:∅⟶𝐵))
41	39, 40	bitr3id 285	. . . . 5 ⊢ (𝑓 = ∅ → ((𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓) ↔ ∅:∅⟶𝐵))
42	37, 41	spcev 3569	. . . 4 ⊢ (∅:∅⟶𝐵 → ∃𝑓(𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓))
43	36, 42	mp1i 13	. . 3 ⊢ (∀𝑥 ∈ ∅ ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓))
44		ssun1 4137	. . . . . . 7 ⊢ 𝑤 ⊆ (𝑤 ∪ {𝑧})
45		ssralv 4012	. . . . . . 7 ⊢ (𝑤 ⊆ (𝑤 ∪ {𝑧}) → (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦 ∈ 𝐵 𝜑 → ∀𝑥 ∈ 𝑤 ∃𝑦 ∈ 𝐵 𝜑))
46	44, 45	ax-mp 5	. . . . . 6 ⊢ (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦 ∈ 𝐵 𝜑 → ∀𝑥 ∈ 𝑤 ∃𝑦 ∈ 𝐵 𝜑)
47	46	imim1i 63	. . . . 5 ⊢ ((∀𝑥 ∈ 𝑤 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)))
48		ssun2 4138	. . . . . . . . 9 ⊢ {𝑧} ⊆ (𝑤 ∪ {𝑧})
49		ssralv 4012	. . . . . . . . 9 ⊢ ({𝑧} ⊆ (𝑤 ∪ {𝑧}) → (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦 ∈ 𝐵 𝜑 → ∀𝑥 ∈ {𝑧}∃𝑦 ∈ 𝐵 𝜑))
50	48, 49	ax-mp 5	. . . . . . . 8 ⊢ (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦 ∈ 𝐵 𝜑 → ∀𝑥 ∈ {𝑧}∃𝑦 ∈ 𝐵 𝜑)
51		ralsnsg 4630	. . . . . . . . . 10 ⊢ (𝑧 ∈ V → (∀𝑥 ∈ {𝑧}∃𝑦 ∈ 𝐵 𝜑 ↔ [𝑧 / 𝑥]∃𝑦 ∈ 𝐵 𝜑))
52	51	elv 3449	. . . . . . . . 9 ⊢ (∀𝑥 ∈ {𝑧}∃𝑦 ∈ 𝐵 𝜑 ↔ [𝑧 / 𝑥]∃𝑦 ∈ 𝐵 𝜑)
53		sbcrex 3835	. . . . . . . . 9 ⊢ ([𝑧 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
54	52, 53	bitri 275	. . . . . . . 8 ⊢ (∀𝑥 ∈ {𝑧}∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
55	50, 54	sylib 218	. . . . . . 7 ⊢ (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
56		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑦 ¬ 𝑧 ∈ 𝑤
57		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑦∃𝑓(𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)
58		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝑔:(𝑤 ∪ {𝑧})⟶𝐵
59		nfcv 2891	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦(𝑤 ∪ {𝑧})
60		nfsbc1v 3770	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦[(𝑔‘𝑥) / 𝑦]𝜑
61	59, 60	nfralw 3283	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑
62	58, 61	nfan 1899	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑)
63	62	nfex 2323	. . . . . . . . 9 ⊢ Ⅎ𝑦∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑)
64	57, 63	nfim 1896	. . . . . . . 8 ⊢ Ⅎ𝑦(∃𝑓(𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑))
65		simprl 770	. . . . . . . . . . . . 13 ⊢ (((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → 𝑓:𝑤⟶𝐵)
66		vex 3448	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ V
67		vex 3448	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
68	66, 67	f1osn 6822	. . . . . . . . . . . . . . 15 ⊢ {⟨𝑧, 𝑦⟩}:{𝑧}–1-1-onto→{𝑦}
69		f1of 6782	. . . . . . . . . . . . . . 15 ⊢ ({⟨𝑧, 𝑦⟩}:{𝑧}–1-1-onto→{𝑦} → {⟨𝑧, 𝑦⟩}:{𝑧}⟶{𝑦})
70	68, 69	mp1i 13	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → {⟨𝑧, 𝑦⟩}:{𝑧}⟶{𝑦})
71		simpl2 1193	. . . . . . . . . . . . . . 15 ⊢ (((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → 𝑦 ∈ 𝐵)
72	71	snssd 4769	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → {𝑦} ⊆ 𝐵)
73	70, 72	fssd 6687	. . . . . . . . . . . . 13 ⊢ (((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → {⟨𝑧, 𝑦⟩}:{𝑧}⟶𝐵)
74		simpl1 1192	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → ¬ 𝑧 ∈ 𝑤)
75		disjsn 4671	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑤)
76	74, 75	sylibr 234	. . . . . . . . . . . . 13 ⊢ (((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → (𝑤 ∩ {𝑧}) = ∅)
77	65, 73, 76	fun2d 6706	. . . . . . . . . . . 12 ⊢ (((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → (𝑓 ∪ {⟨𝑧, 𝑦⟩}):(𝑤 ∪ {𝑧})⟶𝐵)
78		simprr 772	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → ∀𝑥 ∈ 𝑤 𝜓)
79		eleq1a 2823	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝑤 → (𝑧 = 𝑥 → 𝑧 ∈ 𝑤))
80	79	necon3bd 2939	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝑤 → (¬ 𝑧 ∈ 𝑤 → 𝑧 ≠ 𝑥))
81	80	impcom 407	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 𝑧 ∈ 𝑤 ∧ 𝑥 ∈ 𝑤) → 𝑧 ≠ 𝑥)
82		fvunsn 7135	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ≠ 𝑥 → ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) = (𝑓‘𝑥))
83		dfsbcq 3752	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) = (𝑓‘𝑥) → ([((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑 ↔ [(𝑓‘𝑥) / 𝑦]𝜑))
84	83, 21	bitr2di 288	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) = (𝑓‘𝑥) → (𝜓 ↔ [((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
85	81, 82, 84	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((¬ 𝑧 ∈ 𝑤 ∧ 𝑥 ∈ 𝑤) → (𝜓 ↔ [((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
86	85	ralbidva 3154	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑧 ∈ 𝑤 → (∀𝑥 ∈ 𝑤 𝜓 ↔ ∀𝑥 ∈ 𝑤 [((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
87	74, 86	syl 17	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → (∀𝑥 ∈ 𝑤 𝜓 ↔ ∀𝑥 ∈ 𝑤 [((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
88	78, 87	mpbid 232	. . . . . . . . . . . . 13 ⊢ (((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → ∀𝑥 ∈ 𝑤 [((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑)
89		simpl3 1194	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → [𝑧 / 𝑥]𝜑)
90		ffun 6673	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∪ {⟨𝑧, 𝑦⟩}):(𝑤 ∪ {𝑧})⟶𝐵 → Fun (𝑓 ∪ {⟨𝑧, 𝑦⟩}))
91		ssun2 4138	. . . . . . . . . . . . . . . . . 18 ⊢ {⟨𝑧, 𝑦⟩} ⊆ (𝑓 ∪ {⟨𝑧, 𝑦⟩})
92		vsnid 4623	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑧 ∈ {𝑧}
93	67	dmsnop 6177	. . . . . . . . . . . . . . . . . . 19 ⊢ dom {⟨𝑧, 𝑦⟩} = {𝑧}
94	92, 93	eleqtrri 2827	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ dom {⟨𝑧, 𝑦⟩}
95		funssfv 6861	. . . . . . . . . . . . . . . . . 18 ⊢ ((Fun (𝑓 ∪ {⟨𝑧, 𝑦⟩}) ∧ {⟨𝑧, 𝑦⟩} ⊆ (𝑓 ∪ {⟨𝑧, 𝑦⟩}) ∧ 𝑧 ∈ dom {⟨𝑧, 𝑦⟩}) → ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧) = ({⟨𝑧, 𝑦⟩}‘𝑧))
96	91, 94, 95	mp3an23 1455	. . . . . . . . . . . . . . . . 17 ⊢ (Fun (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧) = ({⟨𝑧, 𝑦⟩}‘𝑧))
97	77, 90, 96	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧) = ({⟨𝑧, 𝑦⟩}‘𝑧))
98	66, 67	fvsn 7137	. . . . . . . . . . . . . . . 16 ⊢ ({⟨𝑧, 𝑦⟩}‘𝑧) = 𝑦
99	97, 98	eqtr2di 2781	. . . . . . . . . . . . . . 15 ⊢ (((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → 𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧))
100		ralsnsg 4630	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ V → (∀𝑥 ∈ {𝑧}𝜑 ↔ [𝑧 / 𝑥]𝜑))
101	100	elv 3449	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ {𝑧}𝜑 ↔ [𝑧 / 𝑥]𝜑)
102		elsni 4602	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ {𝑧} → 𝑥 = 𝑧)
103	102	fveq2d 6844	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ {𝑧} → ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧))
104	103	eqeq2d 2740	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ {𝑧} → (𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) ↔ 𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧)))
105	104	biimparc 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧) ∧ 𝑥 ∈ {𝑧}) → 𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥))
106		sbceq1a 3761	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) → (𝜑 ↔ [((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
107	105, 106	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧) ∧ 𝑥 ∈ {𝑧}) → (𝜑 ↔ [((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
108	107	ralbidva 3154	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧) → (∀𝑥 ∈ {𝑧}𝜑 ↔ ∀𝑥 ∈ {𝑧}[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
109	101, 108	bitr3id 285	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧) → ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥 ∈ {𝑧}[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
110	99, 109	syl 17	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥 ∈ {𝑧}[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
111	89, 110	mpbid 232	. . . . . . . . . . . . 13 ⊢ (((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → ∀𝑥 ∈ {𝑧}[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑)
112		ralun 4157	. . . . . . . . . . . . 13 ⊢ ((∀𝑥 ∈ 𝑤 [((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑 ∧ ∀𝑥 ∈ {𝑧}[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑) → ∀𝑥 ∈ (𝑤 ∪ {𝑧})[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑)
113	88, 111, 112	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → ∀𝑥 ∈ (𝑤 ∪ {𝑧})[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑)
114		vex 3448	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
115		snex 5386	. . . . . . . . . . . . . 14 ⊢ {⟨𝑧, 𝑦⟩} ∈ V
116	114, 115	unex 7700	. . . . . . . . . . . . 13 ⊢ (𝑓 ∪ {⟨𝑧, 𝑦⟩}) ∈ V
117		feq1 6648	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → (𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ↔ (𝑓 ∪ {⟨𝑧, 𝑦⟩}):(𝑤 ∪ {𝑧})⟶𝐵))
118		fveq1 6839	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → (𝑔‘𝑥) = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥))
119	118	sbceq1d 3755	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → ([(𝑔‘𝑥) / 𝑦]𝜑 ↔ [((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
120	119	ralbidv 3156	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → (∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑 ↔ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
121	117, 120	anbi12d 632	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → ((𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑) ↔ ((𝑓 ∪ {⟨𝑧, 𝑦⟩}):(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑)))
122	116, 121	spcev 3569	. . . . . . . . . . . 12 ⊢ (((𝑓 ∪ {⟨𝑧, 𝑦⟩}):(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑))
123	77, 113, 122	syl2anc 584	. . . . . . . . . . 11 ⊢ (((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑))
124	123	ex 412	. . . . . . . . . 10 ⊢ ((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) → ((𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑)))
125	124	exlimdv 1933	. . . . . . . . 9 ⊢ ((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) → (∃𝑓(𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑)))
126	125	3exp 1119	. . . . . . . 8 ⊢ (¬ 𝑧 ∈ 𝑤 → (𝑦 ∈ 𝐵 → ([𝑧 / 𝑥]𝜑 → (∃𝑓(𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑)))))
127	56, 64, 126	rexlimd 3242	. . . . . . 7 ⊢ (¬ 𝑧 ∈ 𝑤 → (∃𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 → (∃𝑓(𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑))))
128	55, 127	syl5 34	. . . . . 6 ⊢ (¬ 𝑧 ∈ 𝑤 → (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦 ∈ 𝐵 𝜑 → (∃𝑓(𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓) → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑))))
129	128	a2d 29	. . . . 5 ⊢ (¬ 𝑧 ∈ 𝑤 → ((∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦 ∈ 𝐵 𝜑 → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑))))
130	47, 129	syl5 34	. . . 4 ⊢ (¬ 𝑧 ∈ 𝑤 → ((∀𝑥 ∈ 𝑤 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦 ∈ 𝐵 𝜑 → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑))))
131	130	adantl 481	. . 3 ⊢ ((𝑤 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑤) → ((∀𝑥 ∈ 𝑤 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦 ∈ 𝐵 𝜑 → ∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑))))
132	6, 12, 29, 35, 43, 131	findcard2s 9106	. 2 ⊢ (𝐴 ∈ Fin → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)))
133	132	imp 406	1 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  Vcvv 3444  [wsbc 3750   ∪ cun 3909   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  {csn 4585  ⟨cop 4591  dom cdm 5631  Fun wfun 6493  ⟶wf 6495  –1-1-onto→wf1o 6498  ‘cfv 6499  Fincfn 8895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-om 7823  df-en 8896  df-fin 8899

This theorem is referenced by:  fissuni  9284  fipreima  9285  indexfi  9287  finacn  9979  axcc4dom  10370  ttukeylem6  10443  firest  17371  ablfaclem3  20003  ablfac2  20005  cmpcovf  23311  cmpsub  23320  tgcmp  23321  hauscmplem  23326  comppfsc  23452  ptcnplem  23541  alexsubALTlem3  23969  alexsubALT  23971  tsmsxplem1  24073  ovolicc2lem5  25455  ovolicc2  25456  limciun  25828  cvmliftlem15  35278  matunitlindflem2  37604  ptrecube  37607  istotbnd3  37758  sstotbnd2  37761  sstotbnd  37762  prdsbnd  37780  prdstotbnd  37781  heiborlem1  37798  heibor  37808  kelac1  43045  hbt  43112