Theorem dfac9 8841

Description: Equivalence of the axiom of choice with a statement related to ac9 9188; definition AC3 of [Schechter] p. 139. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Assertion

Ref	Expression
dfac9	⊢ (CHOICE ↔ ∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅))

Distinct variable group:   𝑥,𝑓

Proof of Theorem dfac9

Dummy variables 𝑔 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfac3 8827	. 2 ⊢ (CHOICE ↔ ∀𝑠∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))
2		vex 3176	. . . . . . 7 ⊢ 𝑓 ∈ V
3	2	rnex 6992	. . . . . 6 ⊢ ran 𝑓 ∈ V
4		raleq 3115	. . . . . . 7 ⊢ (𝑠 = ran 𝑓 → (∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) ↔ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)))
5	4	exbidv 1837	. . . . . 6 ⊢ (𝑠 = ran 𝑓 → (∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) ↔ ∃𝑔∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)))
6	3, 5	spcv 3272	. . . . 5 ⊢ (∀𝑠∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) → ∃𝑔∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))
7		df-nel 2783	. . . . . . . . . . . . . . 15 ⊢ (∅ ∉ ran 𝑓 ↔ ¬ ∅ ∈ ran 𝑓)
8	7	biimpi 205	. . . . . . . . . . . . . 14 ⊢ (∅ ∉ ran 𝑓 → ¬ ∅ ∈ ran 𝑓)
9	8	ad2antlr 759	. . . . . . . . . . . . 13 ⊢ (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ 𝑥 ∈ dom 𝑓) → ¬ ∅ ∈ ran 𝑓)
10		fvelrn 6260	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝑓 ∧ 𝑥 ∈ dom 𝑓) → (𝑓‘𝑥) ∈ ran 𝑓)
11	10	adantlr 747	. . . . . . . . . . . . . . 15 ⊢ (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ 𝑥 ∈ dom 𝑓) → (𝑓‘𝑥) ∈ ran 𝑓)
12		eleq1 2676	. . . . . . . . . . . . . . 15 ⊢ ((𝑓‘𝑥) = ∅ → ((𝑓‘𝑥) ∈ ran 𝑓 ↔ ∅ ∈ ran 𝑓))
13	11, 12	syl5ibcom 234	. . . . . . . . . . . . . 14 ⊢ (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ 𝑥 ∈ dom 𝑓) → ((𝑓‘𝑥) = ∅ → ∅ ∈ ran 𝑓))
14	13	necon3bd 2796	. . . . . . . . . . . . 13 ⊢ (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ 𝑥 ∈ dom 𝑓) → (¬ ∅ ∈ ran 𝑓 → (𝑓‘𝑥) ≠ ∅))
15	9, 14	mpd 15	. . . . . . . . . . . 12 ⊢ (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ 𝑥 ∈ dom 𝑓) → (𝑓‘𝑥) ≠ ∅)
16	15	adantlr 747	. . . . . . . . . . 11 ⊢ ((((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) ∧ 𝑥 ∈ dom 𝑓) → (𝑓‘𝑥) ≠ ∅)
17		simpll 786	. . . . . . . . . . . . 13 ⊢ (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) → Fun 𝑓)
18	17, 10	sylan 487	. . . . . . . . . . . 12 ⊢ ((((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) ∧ 𝑥 ∈ dom 𝑓) → (𝑓‘𝑥) ∈ ran 𝑓)
19		simplr 788	. . . . . . . . . . . 12 ⊢ ((((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) ∧ 𝑥 ∈ dom 𝑓) → ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))
20		neeq1 2844	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (𝑓‘𝑥) → (𝑡 ≠ ∅ ↔ (𝑓‘𝑥) ≠ ∅))
21		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝑓‘𝑥) → (𝑔‘𝑡) = (𝑔‘(𝑓‘𝑥)))
22		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = (𝑓‘𝑥) → 𝑡 = (𝑓‘𝑥))
23	21, 22	eleq12d 2682	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (𝑓‘𝑥) → ((𝑔‘𝑡) ∈ 𝑡 ↔ (𝑔‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥)))
24	20, 23	imbi12d 333	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑓‘𝑥) → ((𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) ↔ ((𝑓‘𝑥) ≠ ∅ → (𝑔‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥))))
25	24	rspcva 3280	. . . . . . . . . . . 12 ⊢ (((𝑓‘𝑥) ∈ ran 𝑓 ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) → ((𝑓‘𝑥) ≠ ∅ → (𝑔‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥)))
26	18, 19, 25	syl2anc 691	. . . . . . . . . . 11 ⊢ ((((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) ∧ 𝑥 ∈ dom 𝑓) → ((𝑓‘𝑥) ≠ ∅ → (𝑔‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥)))
27	16, 26	mpd 15	. . . . . . . . . 10 ⊢ ((((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) ∧ 𝑥 ∈ dom 𝑓) → (𝑔‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥))
28	27	ralrimiva 2949	. . . . . . . . 9 ⊢ (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) → ∀𝑥 ∈ dom 𝑓(𝑔‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥))
29	2	dmex 6991	. . . . . . . . . 10 ⊢ dom 𝑓 ∈ V
30		mptelixpg 7831	. . . . . . . . . 10 ⊢ (dom 𝑓 ∈ V → ((𝑥 ∈ dom 𝑓 ↦ (𝑔‘(𝑓‘𝑥))) ∈ X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ↔ ∀𝑥 ∈ dom 𝑓(𝑔‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥)))
31	29, 30	ax-mp 5	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom 𝑓 ↦ (𝑔‘(𝑓‘𝑥))) ∈ X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ↔ ∀𝑥 ∈ dom 𝑓(𝑔‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥))
32	28, 31	sylibr 223	. . . . . . . 8 ⊢ (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) → (𝑥 ∈ dom 𝑓 ↦ (𝑔‘(𝑓‘𝑥))) ∈ X𝑥 ∈ dom 𝑓(𝑓‘𝑥))
33		ne0i 3880	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝑓 ↦ (𝑔‘(𝑓‘𝑥))) ∈ X𝑥 ∈ dom 𝑓(𝑓‘𝑥) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅)
34	32, 33	syl 17	. . . . . . 7 ⊢ (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅)
35	34	ex 449	. . . . . 6 ⊢ ((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → (∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅))
36	35	exlimdv 1848	. . . . 5 ⊢ ((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → (∃𝑔∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅))
37	6, 36	syl5com 31	. . . 4 ⊢ (∀𝑠∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) → ((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅))
38	37	alrimiv 1842	. . 3 ⊢ (∀𝑠∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) → ∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅))
39		fnresi 5922	. . . . . . 7 ⊢ ( I ↾ (𝑠 ∖ {∅})) Fn (𝑠 ∖ {∅})
40		fnfun 5902	. . . . . . 7 ⊢ (( I ↾ (𝑠 ∖ {∅})) Fn (𝑠 ∖ {∅}) → Fun ( I ↾ (𝑠 ∖ {∅})))
41	39, 40	ax-mp 5	. . . . . 6 ⊢ Fun ( I ↾ (𝑠 ∖ {∅}))
42		neldifsn 4262	. . . . . 6 ⊢ ¬ ∅ ∈ (𝑠 ∖ {∅})
43		vex 3176	. . . . . . . . 9 ⊢ 𝑠 ∈ V
44		difexg 4735	. . . . . . . . 9 ⊢ (𝑠 ∈ V → (𝑠 ∖ {∅}) ∈ V)
45	43, 44	ax-mp 5	. . . . . . . 8 ⊢ (𝑠 ∖ {∅}) ∈ V
46		resiexg 6994	. . . . . . . 8 ⊢ ((𝑠 ∖ {∅}) ∈ V → ( I ↾ (𝑠 ∖ {∅})) ∈ V)
47	45, 46	ax-mp 5	. . . . . . 7 ⊢ ( I ↾ (𝑠 ∖ {∅})) ∈ V
48		funeq 5823	. . . . . . . . 9 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → (Fun 𝑓 ↔ Fun ( I ↾ (𝑠 ∖ {∅}))))
49		rneq 5272	. . . . . . . . . . . . 13 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → ran 𝑓 = ran ( I ↾ (𝑠 ∖ {∅})))
50		rnresi 5398	. . . . . . . . . . . . 13 ⊢ ran ( I ↾ (𝑠 ∖ {∅})) = (𝑠 ∖ {∅})
51	49, 50	syl6eq 2660	. . . . . . . . . . . 12 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → ran 𝑓 = (𝑠 ∖ {∅}))
52	51	eleq2d 2673	. . . . . . . . . . 11 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → (∅ ∈ ran 𝑓 ↔ ∅ ∈ (𝑠 ∖ {∅})))
53	52	notbid 307	. . . . . . . . . 10 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → (¬ ∅ ∈ ran 𝑓 ↔ ¬ ∅ ∈ (𝑠 ∖ {∅})))
54	7, 53	syl5bb 271	. . . . . . . . 9 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → (∅ ∉ ran 𝑓 ↔ ¬ ∅ ∈ (𝑠 ∖ {∅})))
55	48, 54	anbi12d 743	. . . . . . . 8 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → ((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ↔ (Fun ( I ↾ (𝑠 ∖ {∅})) ∧ ¬ ∅ ∈ (𝑠 ∖ {∅}))))
56		dmeq 5246	. . . . . . . . . . . 12 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → dom 𝑓 = dom ( I ↾ (𝑠 ∖ {∅})))
57		dmresi 5376	. . . . . . . . . . . 12 ⊢ dom ( I ↾ (𝑠 ∖ {∅})) = (𝑠 ∖ {∅})
58	56, 57	syl6eq 2660	. . . . . . . . . . 11 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → dom 𝑓 = (𝑠 ∖ {∅}))
59	58	ixpeq1d 7806	. . . . . . . . . 10 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) = X𝑥 ∈ (𝑠 ∖ {∅})(𝑓‘𝑥))
60		fveq1 6102	. . . . . . . . . . . 12 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → (𝑓‘𝑥) = (( I ↾ (𝑠 ∖ {∅}))‘𝑥))
61		fvresi 6344	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑠 ∖ {∅}) → (( I ↾ (𝑠 ∖ {∅}))‘𝑥) = 𝑥)
62	60, 61	sylan9eq 2664	. . . . . . . . . . 11 ⊢ ((𝑓 = ( I ↾ (𝑠 ∖ {∅})) ∧ 𝑥 ∈ (𝑠 ∖ {∅})) → (𝑓‘𝑥) = 𝑥)
63	62	ixpeq2dva 7809	. . . . . . . . . 10 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → X𝑥 ∈ (𝑠 ∖ {∅})(𝑓‘𝑥) = X𝑥 ∈ (𝑠 ∖ {∅})𝑥)
64	59, 63	eqtrd 2644	. . . . . . . . 9 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) = X𝑥 ∈ (𝑠 ∖ {∅})𝑥)
65	64	neeq1d 2841	. . . . . . . 8 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → (X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅ ↔ X𝑥 ∈ (𝑠 ∖ {∅})𝑥 ≠ ∅))
66	55, 65	imbi12d 333	. . . . . . 7 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅) ↔ ((Fun ( I ↾ (𝑠 ∖ {∅})) ∧ ¬ ∅ ∈ (𝑠 ∖ {∅})) → X𝑥 ∈ (𝑠 ∖ {∅})𝑥 ≠ ∅)))
67	47, 66	spcv 3272	. . . . . 6 ⊢ (∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅) → ((Fun ( I ↾ (𝑠 ∖ {∅})) ∧ ¬ ∅ ∈ (𝑠 ∖ {∅})) → X𝑥 ∈ (𝑠 ∖ {∅})𝑥 ≠ ∅))
68	41, 42, 67	mp2ani 710	. . . . 5 ⊢ (∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅) → X𝑥 ∈ (𝑠 ∖ {∅})𝑥 ≠ ∅)
69		n0 3890	. . . . . 6 ⊢ (X𝑥 ∈ (𝑠 ∖ {∅})𝑥 ≠ ∅ ↔ ∃𝑔 𝑔 ∈ X𝑥 ∈ (𝑠 ∖ {∅})𝑥)
70		vex 3176	. . . . . . . . 9 ⊢ 𝑔 ∈ V
71	70	elixp 7801	. . . . . . . 8 ⊢ (𝑔 ∈ X𝑥 ∈ (𝑠 ∖ {∅})𝑥 ↔ (𝑔 Fn (𝑠 ∖ {∅}) ∧ ∀𝑥 ∈ (𝑠 ∖ {∅})(𝑔‘𝑥) ∈ 𝑥))
72		eldifsn 4260	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝑠 ∖ {∅}) ↔ (𝑥 ∈ 𝑠 ∧ 𝑥 ≠ ∅))
73	72	imbi1i 338	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (𝑠 ∖ {∅}) → (𝑔‘𝑥) ∈ 𝑥) ↔ ((𝑥 ∈ 𝑠 ∧ 𝑥 ≠ ∅) → (𝑔‘𝑥) ∈ 𝑥))
74		impexp 461	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝑠 ∧ 𝑥 ≠ ∅) → (𝑔‘𝑥) ∈ 𝑥) ↔ (𝑥 ∈ 𝑠 → (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)))
75	73, 74	bitri 263	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝑠 ∖ {∅}) → (𝑔‘𝑥) ∈ 𝑥) ↔ (𝑥 ∈ 𝑠 → (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)))
76	75	ralbii2 2961	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ (𝑠 ∖ {∅})(𝑔‘𝑥) ∈ 𝑥 ↔ ∀𝑥 ∈ 𝑠 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥))
77		neeq1 2844	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → (𝑥 ≠ ∅ ↔ 𝑡 ≠ ∅))
78		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → (𝑔‘𝑥) = (𝑔‘𝑡))
79		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑡 → 𝑥 = 𝑡)
80	78, 79	eleq12d 2682	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → ((𝑔‘𝑥) ∈ 𝑥 ↔ (𝑔‘𝑡) ∈ 𝑡))
81	77, 80	imbi12d 333	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑡 → ((𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥) ↔ (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)))
82	81	cbvralv 3147	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝑠 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥) ↔ ∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))
83	76, 82	bitri 263	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ (𝑠 ∖ {∅})(𝑔‘𝑥) ∈ 𝑥 ↔ ∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))
84	83	biimpi 205	. . . . . . . . 9 ⊢ (∀𝑥 ∈ (𝑠 ∖ {∅})(𝑔‘𝑥) ∈ 𝑥 → ∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))
85	84	adantl 481	. . . . . . . 8 ⊢ ((𝑔 Fn (𝑠 ∖ {∅}) ∧ ∀𝑥 ∈ (𝑠 ∖ {∅})(𝑔‘𝑥) ∈ 𝑥) → ∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))
86	71, 85	sylbi 206	. . . . . . 7 ⊢ (𝑔 ∈ X𝑥 ∈ (𝑠 ∖ {∅})𝑥 → ∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))
87	86	eximi 1752	. . . . . 6 ⊢ (∃𝑔 𝑔 ∈ X𝑥 ∈ (𝑠 ∖ {∅})𝑥 → ∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))
88	69, 87	sylbi 206	. . . . 5 ⊢ (X𝑥 ∈ (𝑠 ∖ {∅})𝑥 ≠ ∅ → ∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))
89	68, 88	syl 17	. . . 4 ⊢ (∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅) → ∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))
90	89	alrimiv 1842	. . 3 ⊢ (∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅) → ∀𝑠∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))
91	38, 90	impbii 198	. 2 ⊢ (∀𝑠∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) ↔ ∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅))
92	1, 91	bitri 263	1 ⊢ (CHOICE ↔ ∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781  ∀wral 2896  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  {csn 4125   ↦ cmpt 4643   I cid 4948  dom cdm 5038  ran crn 5039   ↾ cres 5040  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804  Xcixp 7794  CHOICEwac 8821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ixp 7795  df-ac 8822

This theorem is referenced by:  dfac14  21231  dfac21  36654