Theorem ac6sfi 9186

Description: A version of ac6s 10396 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 6641	. . . . . 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 1922	. . . 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 6641	. . . . . 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 1922	. . . 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 6641	. . . . . . 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 1922	. . . . 5 ⊢ (𝑢 = (𝑤 ∪ {𝑧}) → (∃𝑓(𝑓:𝑢⟶𝐵 ∧ ∀𝑥 ∈ 𝑢 𝜓) ↔ ∃𝑓(𝑓:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓)))
18		feq1 6640	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑓:(𝑤 ∪ {𝑧})⟶𝐵 ↔ 𝑔:(𝑤 ∪ {𝑧})⟶𝐵))
19		fvex 6847	. . . . . . . . . 10 ⊢ (𝑓‘𝑥) ∈ V
20		ac6sfi.1	. . . . . . . . . 10 ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))
21	19, 20	sbcie 3782	. . . . . . . . 9 ⊢ ([(𝑓‘𝑥) / 𝑦]𝜑 ↔ 𝜓)
22		fveq1 6833	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑥) = (𝑔‘𝑥))
23	22	sbceq1d 3745	. . . . . . . . 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 3159	. . . . . . 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 2038	. . . . 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 6641	. . . . . 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 1922	. . . 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 6715	. . . 4 ⊢ ∅:∅⟶𝐵
37		0ex 5252	. . . . 5 ⊢ ∅ ∈ V
38		ral0 4451	. . . . . . 7 ⊢ ∀𝑥 ∈ ∅ 𝜓
39	38	biantru 529	. . . . . 6 ⊢ (𝑓:∅⟶𝐵 ↔ (𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓))
40		feq1 6640	. . . . . 6 ⊢ (𝑓 = ∅ → (𝑓:∅⟶𝐵 ↔ ∅:∅⟶𝐵))
41	39, 40	bitr3id 285	. . . . 5 ⊢ (𝑓 = ∅ → ((𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓) ↔ ∅:∅⟶𝐵))
42	37, 41	spcev 3560	. . . 4 ⊢ (∅:∅⟶𝐵 → ∃𝑓(𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓))
43	36, 42	mp1i 13	. . 3 ⊢ (∀𝑥 ∈ ∅ ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓))
44		ssun1 4130	. . . . . . 7 ⊢ 𝑤 ⊆ (𝑤 ∪ {𝑧})
45		ssralv 4002	. . . . . . 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 4131	. . . . . . . . 9 ⊢ {𝑧} ⊆ (𝑤 ∪ {𝑧})
49		ssralv 4002	. . . . . . . . 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 4627	. . . . . . . . . 10 ⊢ (𝑧 ∈ V → (∀𝑥 ∈ {𝑧}∃𝑦 ∈ 𝐵 𝜑 ↔ [𝑧 / 𝑥]∃𝑦 ∈ 𝐵 𝜑))
52	51	elv 3445	. . . . . . . . 9 ⊢ (∀𝑥 ∈ {𝑧}∃𝑦 ∈ 𝐵 𝜑 ↔ [𝑧 / 𝑥]∃𝑦 ∈ 𝐵 𝜑)
53		sbcrex 3825	. . . . . . . . 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 1915	. . . . . . . 8 ⊢ Ⅎ𝑦 ¬ 𝑧 ∈ 𝑤
57		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑦∃𝑓(𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)
58		nfv 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝑔:(𝑤 ∪ {𝑧})⟶𝐵
59		nfcv 2898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦(𝑤 ∪ {𝑧})
60		nfsbc1v 3760	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦[(𝑔‘𝑥) / 𝑦]𝜑
61	59, 60	nfralw 3283	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑
62	58, 61	nfan 1900	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑)
63	62	nfex 2329	. . . . . . . . 9 ⊢ Ⅎ𝑦∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑)
64	57, 63	nfim 1897	. . . . . . . 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 3444	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ V
67		vex 3444	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
68	66, 67	f1osn 6815	. . . . . . . . . . . . . . 15 ⊢ {⟨𝑧, 𝑦⟩}:{𝑧}–1-1-onto→{𝑦}
69		f1of 6774	. . . . . . . . . . . . . . 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 4765	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → {𝑦} ⊆ 𝐵)
73	70, 72	fssd 6679	. . . . . . . . . . . . 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 4668	. . . . . . . . . . . . . 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 6698	. . . . . . . . . . . 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 2831	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝑤 → (𝑧 = 𝑥 → 𝑧 ∈ 𝑤))
80	79	necon3bd 2946	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝑤 → (¬ 𝑧 ∈ 𝑤 → 𝑧 ≠ 𝑥))
81	80	impcom 407	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 𝑧 ∈ 𝑤 ∧ 𝑥 ∈ 𝑤) → 𝑧 ≠ 𝑥)
82		fvunsn 7125	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ≠ 𝑥 → ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) = (𝑓‘𝑥))
83		dfsbcq 3742	. . . . . . . . . . . . . . . . . 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 3157	. . . . . . . . . . . . . . 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 6665	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∪ {⟨𝑧, 𝑦⟩}):(𝑤 ∪ {𝑧})⟶𝐵 → Fun (𝑓 ∪ {⟨𝑧, 𝑦⟩}))
91		ssun2 4131	. . . . . . . . . . . . . . . . . 18 ⊢ {⟨𝑧, 𝑦⟩} ⊆ (𝑓 ∪ {⟨𝑧, 𝑦⟩})
92		vsnid 4620	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑧 ∈ {𝑧}
93	67	dmsnop 6174	. . . . . . . . . . . . . . . . . . 19 ⊢ dom {⟨𝑧, 𝑦⟩} = {𝑧}
94	92, 93	eleqtrri 2835	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ dom {⟨𝑧, 𝑦⟩}
95		funssfv 6855	. . . . . . . . . . . . . . . . . 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 7127	. . . . . . . . . . . . . . . 16 ⊢ ({⟨𝑧, 𝑦⟩}‘𝑧) = 𝑦
99	97, 98	eqtr2di 2788	. . . . . . . . . . . . . . 15 ⊢ (((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → 𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧))
100		ralsnsg 4627	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ V → (∀𝑥 ∈ {𝑧}𝜑 ↔ [𝑧 / 𝑥]𝜑))
101	100	elv 3445	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ {𝑧}𝜑 ↔ [𝑧 / 𝑥]𝜑)
102		elsni 4597	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ {𝑧} → 𝑥 = 𝑧)
103	102	fveq2d 6838	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ {𝑧} → ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧))
104	103	eqeq2d 2747	. . . . . . . . . . . . . . . . . . 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 3751	. . . . . . . . . . . . . . . . . 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 3157	. . . . . . . . . . . . . . . 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 4150	. . . . . . . . . . . . 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 3444	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
115		snex 5381	. . . . . . . . . . . . . 14 ⊢ {⟨𝑧, 𝑦⟩} ∈ V
116	114, 115	unex 7689	. . . . . . . . . . . . 13 ⊢ (𝑓 ∪ {⟨𝑧, 𝑦⟩}) ∈ V
117		feq1 6640	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → (𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ↔ (𝑓 ∪ {⟨𝑧, 𝑦⟩}):(𝑤 ∪ {𝑧})⟶𝐵))
118		fveq1 6833	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → (𝑔‘𝑥) = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥))
119	118	sbceq1d 3745	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → ([(𝑔‘𝑥) / 𝑦]𝜑 ↔ [((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
120	119	ralbidv 3159	. . . . . . . . . . . . . 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 3560	. . . . . . . . . . . 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 1934	. . . . . . . . 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 3243	. . . . . . 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 9092	. 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 1541  ∃wex 1780   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  Vcvv 3440  [wsbc 3740   ∪ cun 3899   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  {csn 4580  ⟨cop 4586  dom cdm 5624  Fun wfun 6486  ⟶wf 6488  –1-1-onto→wf1o 6491  ‘cfv 6492  Fincfn 8885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7809  df-en 8886  df-fin 8889

This theorem is referenced by:  fissuni  9259  fipreima  9260  indexfi  9262  finacn  9962  axcc4dom  10353  ttukeylem6  10426  firest  17354  ablfaclem3  20020  ablfac2  20022  cmpcovf  23337  cmpsub  23346  tgcmp  23347  hauscmplem  23352  comppfsc  23478  ptcnplem  23567  alexsubALTlem3  23995  alexsubALT  23997  tsmsxplem1  24099  ovolicc2lem5  25480  ovolicc2  25481  limciun  25853  cvmliftlem15  35494  matunitlindflem2  37820  ptrecube  37823  istotbnd3  37974  sstotbnd2  37977  sstotbnd  37978  prdsbnd  37996  prdstotbnd  37997  heiborlem1  38014  heibor  38024  kelac1  43326  hbt  43393