Theorem ac6sfi 9284

Description: A version of ac6s 10476 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 3323	. . . 4 ⊢ (𝑢 = ∅ → (∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ ∅ ∃𝑦 ∈ 𝐵 𝜑))
2		feq2 6697	. . . . . 6 ⊢ (𝑢 = ∅ → (𝑓:𝑢⟶𝐵 ↔ 𝑓:∅⟶𝐵))
3		raleq 3323	. . . . . 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 1925	. . . 4 ⊢ (𝑢 = ∅ → (∃𝑓(𝑓:𝑢⟶𝐵 ∧ ∀𝑥 ∈ 𝑢 𝜓) ↔ ∃𝑓(𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓)))
6	1, 5	imbi12d 345	. . 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 3323	. . . 4 ⊢ (𝑢 = 𝑤 → (∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝑤 ∃𝑦 ∈ 𝐵 𝜑))
8		feq2 6697	. . . . . 6 ⊢ (𝑢 = 𝑤 → (𝑓:𝑢⟶𝐵 ↔ 𝑓:𝑤⟶𝐵))
9		raleq 3323	. . . . . 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 1925	. . . 4 ⊢ (𝑢 = 𝑤 → (∃𝑓(𝑓:𝑢⟶𝐵 ∧ ∀𝑥 ∈ 𝑢 𝜓) ↔ ∃𝑓(𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)))
12	7, 11	imbi12d 345	. . 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 3323	. . . 4 ⊢ (𝑢 = (𝑤 ∪ {𝑧}) → (∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦 ∈ 𝐵 𝜑))
14		feq2 6697	. . . . . . 7 ⊢ (𝑢 = (𝑤 ∪ {𝑧}) → (𝑓:𝑢⟶𝐵 ↔ 𝑓:(𝑤 ∪ {𝑧})⟶𝐵))
15		raleq 3323	. . . . . . 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 1925	. . . . 5 ⊢ (𝑢 = (𝑤 ∪ {𝑧}) → (∃𝑓(𝑓:𝑢⟶𝐵 ∧ ∀𝑥 ∈ 𝑢 𝜓) ↔ ∃𝑓(𝑓:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})𝜓)))
18		feq1 6696	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝑓:(𝑤 ∪ {𝑧})⟶𝐵 ↔ 𝑔:(𝑤 ∪ {𝑧})⟶𝐵))
19		fvex 6902	. . . . . . . . . 10 ⊢ (𝑓‘𝑥) ∈ V
20		ac6sfi.1	. . . . . . . . . 10 ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))
21	19, 20	sbcie 3820	. . . . . . . . 9 ⊢ ([(𝑓‘𝑥) / 𝑦]𝜑 ↔ 𝜓)
22		fveq1 6888	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑥) = (𝑔‘𝑥))
23	22	sbceq1d 3782	. . . . . . . . 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 3178	. . . . . . 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 2041	. . . . 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 345	. . 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 3323	. . . 4 ⊢ (𝑢 = 𝐴 → (∀𝑥 ∈ 𝑢 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑))
31		feq2 6697	. . . . . 6 ⊢ (𝑢 = 𝐴 → (𝑓:𝑢⟶𝐵 ↔ 𝑓:𝐴⟶𝐵))
32		raleq 3323	. . . . . 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 1925	. . . 4 ⊢ (𝑢 = 𝐴 → (∃𝑓(𝑓:𝑢⟶𝐵 ∧ ∀𝑥 ∈ 𝑢 𝜓) ↔ ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)))
35	30, 34	imbi12d 345	. . 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 6770	. . . 4 ⊢ ∅:∅⟶𝐵
37		0ex 5307	. . . . 5 ⊢ ∅ ∈ V
38		ral0 4512	. . . . . . 7 ⊢ ∀𝑥 ∈ ∅ 𝜓
39	38	biantru 531	. . . . . 6 ⊢ (𝑓:∅⟶𝐵 ↔ (𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓))
40		feq1 6696	. . . . . 6 ⊢ (𝑓 = ∅ → (𝑓:∅⟶𝐵 ↔ ∅:∅⟶𝐵))
41	39, 40	bitr3id 285	. . . . 5 ⊢ (𝑓 = ∅ → ((𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓) ↔ ∅:∅⟶𝐵))
42	37, 41	spcev 3597	. . . 4 ⊢ (∅:∅⟶𝐵 → ∃𝑓(𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓))
43	36, 42	mp1i 13	. . 3 ⊢ (∀𝑥 ∈ ∅ ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:∅⟶𝐵 ∧ ∀𝑥 ∈ ∅ 𝜓))
44		ssun1 4172	. . . . . . 7 ⊢ 𝑤 ⊆ (𝑤 ∪ {𝑧})
45		ssralv 4050	. . . . . . 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 4173	. . . . . . . . 9 ⊢ {𝑧} ⊆ (𝑤 ∪ {𝑧})
49		ssralv 4050	. . . . . . . . 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 4672	. . . . . . . . . 10 ⊢ (𝑧 ∈ V → (∀𝑥 ∈ {𝑧}∃𝑦 ∈ 𝐵 𝜑 ↔ [𝑧 / 𝑥]∃𝑦 ∈ 𝐵 𝜑))
52	51	elv 3481	. . . . . . . . 9 ⊢ (∀𝑥 ∈ {𝑧}∃𝑦 ∈ 𝐵 𝜑 ↔ [𝑧 / 𝑥]∃𝑦 ∈ 𝐵 𝜑)
53		sbcrex 3869	. . . . . . . . 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 217	. . . . . . 7 ⊢ (∀𝑥 ∈ (𝑤 ∪ {𝑧})∃𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
56		nfv 1918	. . . . . . . 8 ⊢ Ⅎ𝑦 ¬ 𝑧 ∈ 𝑤
57		nfv 1918	. . . . . . . . 9 ⊢ Ⅎ𝑦∃𝑓(𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)
58		nfv 1918	. . . . . . . . . . 11 ⊢ Ⅎ𝑦 𝑔:(𝑤 ∪ {𝑧})⟶𝐵
59		nfcv 2904	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦(𝑤 ∪ {𝑧})
60		nfsbc1v 3797	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦[(𝑔‘𝑥) / 𝑦]𝜑
61	59, 60	nfralw 3309	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑
62	58, 61	nfan 1903	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑)
63	62	nfex 2318	. . . . . . . . 9 ⊢ Ⅎ𝑦∃𝑔(𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ∧ ∀𝑥 ∈ (𝑤 ∪ {𝑧})[(𝑔‘𝑥) / 𝑦]𝜑)
64	57, 63	nfim 1900	. . . . . . . 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 3479	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ V
67		vex 3479	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
68	66, 67	f1osn 6871	. . . . . . . . . . . . . . 15 ⊢ {⟨𝑧, 𝑦⟩}:{𝑧}–1-1-onto→{𝑦}
69		f1of 6831	. . . . . . . . . . . . . . 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 4812	. . . . . . . . . . . . . 14 ⊢ (((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → {𝑦} ⊆ 𝐵)
73	70, 72	fssd 6733	. . . . . . . . . . . . 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 4715	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑤)
76	74, 75	sylibr 233	. . . . . . . . . . . . 13 ⊢ (((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → (𝑤 ∩ {𝑧}) = ∅)
77	65, 73, 76	fun2d 6753	. . . . . . . . . . . 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 2829	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ 𝑤 → (𝑧 = 𝑥 → 𝑧 ∈ 𝑤))
80	79	necon3bd 2955	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝑤 → (¬ 𝑧 ∈ 𝑤 → 𝑧 ≠ 𝑥))
81	80	impcom 409	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ 𝑧 ∈ 𝑤 ∧ 𝑥 ∈ 𝑤) → 𝑧 ≠ 𝑥)
82		fvunsn 7174	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ≠ 𝑥 → ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) = (𝑓‘𝑥))
83		dfsbcq 3779	. . . . . . . . . . . . . . . . . 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 3176	. . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . 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 6718	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∪ {⟨𝑧, 𝑦⟩}):(𝑤 ∪ {𝑧})⟶𝐵 → Fun (𝑓 ∪ {⟨𝑧, 𝑦⟩}))
91		ssun2 4173	. . . . . . . . . . . . . . . . . 18 ⊢ {⟨𝑧, 𝑦⟩} ⊆ (𝑓 ∪ {⟨𝑧, 𝑦⟩})
92		vsnid 4665	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑧 ∈ {𝑧}
93	67	dmsnop 6213	. . . . . . . . . . . . . . . . . . 19 ⊢ dom {⟨𝑧, 𝑦⟩} = {𝑧}
94	92, 93	eleqtrri 2833	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ dom {⟨𝑧, 𝑦⟩}
95		funssfv 6910	. . . . . . . . . . . . . . . . . 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 1454	. . . . . . . . . . . . . . . . 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 7176	. . . . . . . . . . . . . . . 16 ⊢ ({⟨𝑧, 𝑦⟩}‘𝑧) = 𝑦
99	97, 98	eqtr2di 2790	. . . . . . . . . . . . . . 15 ⊢ (((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → 𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧))
100		ralsnsg 4672	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ V → (∀𝑥 ∈ {𝑧}𝜑 ↔ [𝑧 / 𝑥]𝜑))
101	100	elv 3481	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑥 ∈ {𝑧}𝜑 ↔ [𝑧 / 𝑥]𝜑)
102		elsni 4645	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ {𝑧} → 𝑥 = 𝑧)
103	102	fveq2d 6893	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ {𝑧} → ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧))
104	103	eqeq2d 2744	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ {𝑧} → (𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) ↔ 𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧)))
105	104	biimparc 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑧) ∧ 𝑥 ∈ {𝑧}) → 𝑦 = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥))
106		sbceq1a 3788	. . . . . . . . . . . . . . . . . 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 3176	. . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . 13 ⊢ (((¬ 𝑧 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵 ∧ [𝑧 / 𝑥]𝜑) ∧ (𝑓:𝑤⟶𝐵 ∧ ∀𝑥 ∈ 𝑤 𝜓)) → ∀𝑥 ∈ {𝑧}[((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑)
112		ralun 4192	. . . . . . . . . . . . 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 585	. . . . . . . . . . . 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 3479	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
115		snex 5431	. . . . . . . . . . . . . 14 ⊢ {⟨𝑧, 𝑦⟩} ∈ V
116	114, 115	unex 7730	. . . . . . . . . . . . 13 ⊢ (𝑓 ∪ {⟨𝑧, 𝑦⟩}) ∈ V
117		feq1 6696	. . . . . . . . . . . . . 14 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → (𝑔:(𝑤 ∪ {𝑧})⟶𝐵 ↔ (𝑓 ∪ {⟨𝑧, 𝑦⟩}):(𝑤 ∪ {𝑧})⟶𝐵))
118		fveq1 6888	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → (𝑔‘𝑥) = ((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥))
119	118	sbceq1d 3782	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑦⟩}) → ([(𝑔‘𝑥) / 𝑦]𝜑 ↔ [((𝑓 ∪ {⟨𝑧, 𝑦⟩})‘𝑥) / 𝑦]𝜑))
120	119	ralbidv 3178	. . . . . . . . . . . . . 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 3597	. . . . . . . . . . . 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 585	. . . . . . . . . . 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 414	. . . . . . . . . 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 1937	. . . . . . . . 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 1120	. . . . . . . 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 3264	. . . . . . 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 483	. . 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 9162	. 2 ⊢ (𝐴 ∈ Fin → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓)))
133	132	imp 408	1 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  ∃wex 1782   ∈ wcel 2107   ≠ wne 2941  ∀wral 3062  ∃wrex 3071  Vcvv 3475  [wsbc 3777   ∪ cun 3946   ∩ cin 3947   ⊆ wss 3948  ∅c0 4322  {csn 4628  ⟨cop 4634  dom cdm 5676  Fun wfun 6535  ⟶wf 6537  –1-1-onto→wf1o 6540  ‘cfv 6541  Fincfn 8936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-om 7853  df-en 8937  df-fin 8940

This theorem is referenced by:  fissuni  9354  fipreima  9355  indexfi  9357  finacn  10042  axcc4dom  10433  ttukeylem6  10506  firest  17375  ablfaclem3  19952  ablfac2  19954  cmpcovf  22887  cmpsub  22896  tgcmp  22897  hauscmplem  22902  comppfsc  23028  ptcnplem  23117  alexsubALTlem3  23545  alexsubALT  23547  tsmsxplem1  23649  ovolicc2lem5  25030  ovolicc2  25031  limciun  25403  cvmliftlem15  34278  matunitlindflem2  36474  ptrecube  36477  istotbnd3  36628  sstotbnd2  36631  sstotbnd  36632  prdsbnd  36650  prdstotbnd  36651  heiborlem1  36668  heibor  36678  kelac1  41791  hbt  41858