Theorem dfac9 10179

Description: Equivalence of the axiom of choice with a statement related to ac9 10526; 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 10164	. 2 ⊢ (CHOICE ↔ ∀𝑠∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))
2		vex 3466	. . . . . . 7 ⊢ 𝑓 ∈ V
3	2	rnex 7923	. . . . . 6 ⊢ ran 𝑓 ∈ V
4		raleq 3312	. . . . . . 7 ⊢ (𝑠 = ran 𝑓 → (∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) ↔ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)))
5	4	exbidv 1917	. . . . . 6 ⊢ (𝑠 = ran 𝑓 → (∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) ↔ ∃𝑔∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)))
6	3, 5	spcv 3591	. . . . 5 ⊢ (∀𝑠∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) → ∃𝑔∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))
7		df-nel 3037	. . . . . . . . . . . . . . 15 ⊢ (∅ ∉ ran 𝑓 ↔ ¬ ∅ ∈ ran 𝑓)
8	7	biimpi 215	. . . . . . . . . . . . . 14 ⊢ (∅ ∉ ran 𝑓 → ¬ ∅ ∈ ran 𝑓)
9	8	ad2antlr 725	. . . . . . . . . . . . 13 ⊢ (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ 𝑥 ∈ dom 𝑓) → ¬ ∅ ∈ ran 𝑓)
10		fvelrn 7090	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝑓 ∧ 𝑥 ∈ dom 𝑓) → (𝑓‘𝑥) ∈ ran 𝑓)
11	10	adantlr 713	. . . . . . . . . . . . . . 15 ⊢ (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ 𝑥 ∈ dom 𝑓) → (𝑓‘𝑥) ∈ ran 𝑓)
12		eleq1 2814	. . . . . . . . . . . . . . 15 ⊢ ((𝑓‘𝑥) = ∅ → ((𝑓‘𝑥) ∈ ran 𝑓 ↔ ∅ ∈ ran 𝑓))
13	11, 12	syl5ibcom 244	. . . . . . . . . . . . . 14 ⊢ (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ 𝑥 ∈ dom 𝑓) → ((𝑓‘𝑥) = ∅ → ∅ ∈ ran 𝑓))
14	13	necon3bd 2944	. . . . . . . . . . . . 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 713	. . . . . . . . . . 11 ⊢ ((((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) ∧ 𝑥 ∈ dom 𝑓) → (𝑓‘𝑥) ≠ ∅)
17		neeq1 2993	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑓‘𝑥) → (𝑡 ≠ ∅ ↔ (𝑓‘𝑥) ≠ ∅))
18		fveq2 6901	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (𝑓‘𝑥) → (𝑔‘𝑡) = (𝑔‘(𝑓‘𝑥)))
19		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑡 = (𝑓‘𝑥) → 𝑡 = (𝑓‘𝑥))
20	18, 19	eleq12d 2820	. . . . . . . . . . . . 13 ⊢ (𝑡 = (𝑓‘𝑥) → ((𝑔‘𝑡) ∈ 𝑡 ↔ (𝑔‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥)))
21	17, 20	imbi12d 343	. . . . . . . . . . . 12 ⊢ (𝑡 = (𝑓‘𝑥) → ((𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) ↔ ((𝑓‘𝑥) ≠ ∅ → (𝑔‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥))))
22		simplr 767	. . . . . . . . . . . 12 ⊢ ((((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) ∧ 𝑥 ∈ dom 𝑓) → ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))
23	10	ad4ant14 750	. . . . . . . . . . . 12 ⊢ ((((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) ∧ 𝑥 ∈ dom 𝑓) → (𝑓‘𝑥) ∈ ran 𝑓)
24	21, 22, 23	rspcdva 3609	. . . . . . . . . . 11 ⊢ ((((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) ∧ 𝑥 ∈ dom 𝑓) → ((𝑓‘𝑥) ≠ ∅ → (𝑔‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥)))
25	16, 24	mpd 15	. . . . . . . . . 10 ⊢ ((((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) ∧ 𝑥 ∈ dom 𝑓) → (𝑔‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥))
26	25	ralrimiva 3136	. . . . . . . . 9 ⊢ (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) → ∀𝑥 ∈ dom 𝑓(𝑔‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥))
27	2	dmex 7922	. . . . . . . . . 10 ⊢ dom 𝑓 ∈ V
28		mptelixpg 8964	. . . . . . . . . 10 ⊢ (dom 𝑓 ∈ V → ((𝑥 ∈ dom 𝑓 ↦ (𝑔‘(𝑓‘𝑥))) ∈ X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ↔ ∀𝑥 ∈ dom 𝑓(𝑔‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥)))
29	27, 28	ax-mp 5	. . . . . . . . 9 ⊢ ((𝑥 ∈ dom 𝑓 ↦ (𝑔‘(𝑓‘𝑥))) ∈ X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ↔ ∀𝑥 ∈ dom 𝑓(𝑔‘(𝑓‘𝑥)) ∈ (𝑓‘𝑥))
30	26, 29	sylibr 233	. . . . . . . 8 ⊢ (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) → (𝑥 ∈ dom 𝑓 ↦ (𝑔‘(𝑓‘𝑥))) ∈ X𝑥 ∈ dom 𝑓(𝑓‘𝑥))
31	30	ne0d 4338	. . . . . . 7 ⊢ (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ∧ ∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅)
32	31	ex 411	. . . . . 6 ⊢ ((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → (∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅))
33	32	exlimdv 1929	. . . . 5 ⊢ ((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → (∃𝑔∀𝑡 ∈ ran 𝑓(𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅))
34	6, 33	syl5com 31	. . . 4 ⊢ (∀𝑠∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) → ((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅))
35	34	alrimiv 1923	. . 3 ⊢ (∀𝑠∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) → ∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅))
36		fnresi 6690	. . . . . . 7 ⊢ ( I ↾ (𝑠 ∖ {∅})) Fn (𝑠 ∖ {∅})
37		fnfun 6660	. . . . . . 7 ⊢ (( I ↾ (𝑠 ∖ {∅})) Fn (𝑠 ∖ {∅}) → Fun ( I ↾ (𝑠 ∖ {∅})))
38	36, 37	ax-mp 5	. . . . . 6 ⊢ Fun ( I ↾ (𝑠 ∖ {∅}))
39		neldifsn 4801	. . . . . 6 ⊢ ¬ ∅ ∈ (𝑠 ∖ {∅})
40		vex 3466	. . . . . . . . 9 ⊢ 𝑠 ∈ V
41	40	difexi 5335	. . . . . . . 8 ⊢ (𝑠 ∖ {∅}) ∈ V
42		resiexg 7925	. . . . . . . 8 ⊢ ((𝑠 ∖ {∅}) ∈ V → ( I ↾ (𝑠 ∖ {∅})) ∈ V)
43	41, 42	ax-mp 5	. . . . . . 7 ⊢ ( I ↾ (𝑠 ∖ {∅})) ∈ V
44		funeq 6579	. . . . . . . . 9 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → (Fun 𝑓 ↔ Fun ( I ↾ (𝑠 ∖ {∅}))))
45		rneq 5942	. . . . . . . . . . . . 13 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → ran 𝑓 = ran ( I ↾ (𝑠 ∖ {∅})))
46		rnresi 6084	. . . . . . . . . . . . 13 ⊢ ran ( I ↾ (𝑠 ∖ {∅})) = (𝑠 ∖ {∅})
47	45, 46	eqtrdi 2782	. . . . . . . . . . . 12 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → ran 𝑓 = (𝑠 ∖ {∅}))
48	47	eleq2d 2812	. . . . . . . . . . 11 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → (∅ ∈ ran 𝑓 ↔ ∅ ∈ (𝑠 ∖ {∅})))
49	48	notbid 317	. . . . . . . . . 10 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → (¬ ∅ ∈ ran 𝑓 ↔ ¬ ∅ ∈ (𝑠 ∖ {∅})))
50	7, 49	bitrid 282	. . . . . . . . 9 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → (∅ ∉ ran 𝑓 ↔ ¬ ∅ ∈ (𝑠 ∖ {∅})))
51	44, 50	anbi12d 630	. . . . . . . 8 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → ((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) ↔ (Fun ( I ↾ (𝑠 ∖ {∅})) ∧ ¬ ∅ ∈ (𝑠 ∖ {∅}))))
52		dmeq 5910	. . . . . . . . . . . 12 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → dom 𝑓 = dom ( I ↾ (𝑠 ∖ {∅})))
53		dmresi 6061	. . . . . . . . . . . 12 ⊢ dom ( I ↾ (𝑠 ∖ {∅})) = (𝑠 ∖ {∅})
54	52, 53	eqtrdi 2782	. . . . . . . . . . 11 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → dom 𝑓 = (𝑠 ∖ {∅}))
55	54	ixpeq1d 8938	. . . . . . . . . 10 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) = X𝑥 ∈ (𝑠 ∖ {∅})(𝑓‘𝑥))
56		fveq1 6900	. . . . . . . . . . . 12 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → (𝑓‘𝑥) = (( I ↾ (𝑠 ∖ {∅}))‘𝑥))
57		fvresi 7187	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑠 ∖ {∅}) → (( I ↾ (𝑠 ∖ {∅}))‘𝑥) = 𝑥)
58	56, 57	sylan9eq 2786	. . . . . . . . . . 11 ⊢ ((𝑓 = ( I ↾ (𝑠 ∖ {∅})) ∧ 𝑥 ∈ (𝑠 ∖ {∅})) → (𝑓‘𝑥) = 𝑥)
59	58	ixpeq2dva 8941	. . . . . . . . . 10 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → X𝑥 ∈ (𝑠 ∖ {∅})(𝑓‘𝑥) = X𝑥 ∈ (𝑠 ∖ {∅})𝑥)
60	55, 59	eqtrd 2766	. . . . . . . . 9 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) = X𝑥 ∈ (𝑠 ∖ {∅})𝑥)
61	60	neeq1d 2990	. . . . . . . 8 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → (X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅ ↔ X𝑥 ∈ (𝑠 ∖ {∅})𝑥 ≠ ∅))
62	51, 61	imbi12d 343	. . . . . . 7 ⊢ (𝑓 = ( I ↾ (𝑠 ∖ {∅})) → (((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅) ↔ ((Fun ( I ↾ (𝑠 ∖ {∅})) ∧ ¬ ∅ ∈ (𝑠 ∖ {∅})) → X𝑥 ∈ (𝑠 ∖ {∅})𝑥 ≠ ∅)))
63	43, 62	spcv 3591	. . . . . 6 ⊢ (∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅) → ((Fun ( I ↾ (𝑠 ∖ {∅})) ∧ ¬ ∅ ∈ (𝑠 ∖ {∅})) → X𝑥 ∈ (𝑠 ∖ {∅})𝑥 ≠ ∅))
64	38, 39, 63	mp2ani 696	. . . . 5 ⊢ (∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅) → X𝑥 ∈ (𝑠 ∖ {∅})𝑥 ≠ ∅)
65		n0 4349	. . . . . 6 ⊢ (X𝑥 ∈ (𝑠 ∖ {∅})𝑥 ≠ ∅ ↔ ∃𝑔 𝑔 ∈ X𝑥 ∈ (𝑠 ∖ {∅})𝑥)
66		vex 3466	. . . . . . . . 9 ⊢ 𝑔 ∈ V
67	66	elixp 8933	. . . . . . . 8 ⊢ (𝑔 ∈ X𝑥 ∈ (𝑠 ∖ {∅})𝑥 ↔ (𝑔 Fn (𝑠 ∖ {∅}) ∧ ∀𝑥 ∈ (𝑠 ∖ {∅})(𝑔‘𝑥) ∈ 𝑥))
68		eldifsn 4795	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑠 ∖ {∅}) ↔ (𝑥 ∈ 𝑠 ∧ 𝑥 ≠ ∅))
69	68	imbi1i 348	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (𝑠 ∖ {∅}) → (𝑔‘𝑥) ∈ 𝑥) ↔ ((𝑥 ∈ 𝑠 ∧ 𝑥 ≠ ∅) → (𝑔‘𝑥) ∈ 𝑥))
70		impexp 449	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝑠 ∧ 𝑥 ≠ ∅) → (𝑔‘𝑥) ∈ 𝑥) ↔ (𝑥 ∈ 𝑠 → (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)))
71	69, 70	bitri 274	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (𝑠 ∖ {∅}) → (𝑔‘𝑥) ∈ 𝑥) ↔ (𝑥 ∈ 𝑠 → (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥)))
72	71	ralbii2 3079	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ (𝑠 ∖ {∅})(𝑔‘𝑥) ∈ 𝑥 ↔ ∀𝑥 ∈ 𝑠 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥))
73		neeq1 2993	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑡 → (𝑥 ≠ ∅ ↔ 𝑡 ≠ ∅))
74		fveq2 6901	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → (𝑔‘𝑥) = (𝑔‘𝑡))
75		id 22	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑡 → 𝑥 = 𝑡)
76	74, 75	eleq12d 2820	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑡 → ((𝑔‘𝑥) ∈ 𝑥 ↔ (𝑔‘𝑡) ∈ 𝑡))
77	73, 76	imbi12d 343	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑡 → ((𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥) ↔ (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡)))
78	77	cbvralvw 3225	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑠 (𝑥 ≠ ∅ → (𝑔‘𝑥) ∈ 𝑥) ↔ ∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))
79	72, 78	bitri 274	. . . . . . . . 9 ⊢ (∀𝑥 ∈ (𝑠 ∖ {∅})(𝑔‘𝑥) ∈ 𝑥 ↔ ∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))
80	79	biimpi 215	. . . . . . . 8 ⊢ (∀𝑥 ∈ (𝑠 ∖ {∅})(𝑔‘𝑥) ∈ 𝑥 → ∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))
81	67, 80	simplbiim 503	. . . . . . 7 ⊢ (𝑔 ∈ X𝑥 ∈ (𝑠 ∖ {∅})𝑥 → ∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))
82	81	eximi 1830	. . . . . 6 ⊢ (∃𝑔 𝑔 ∈ X𝑥 ∈ (𝑠 ∖ {∅})𝑥 → ∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))
83	65, 82	sylbi 216	. . . . 5 ⊢ (X𝑥 ∈ (𝑠 ∖ {∅})𝑥 ≠ ∅ → ∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))
84	64, 83	syl 17	. . . 4 ⊢ (∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅) → ∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))
85	84	alrimiv 1923	. . 3 ⊢ (∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅) → ∀𝑠∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡))
86	35, 85	impbii 208	. 2 ⊢ (∀𝑠∃𝑔∀𝑡 ∈ 𝑠 (𝑡 ≠ ∅ → (𝑔‘𝑡) ∈ 𝑡) ↔ ∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅))
87	1, 86	bitri 274	1 ⊢ (CHOICE ↔ ∀𝑓((Fun 𝑓 ∧ ∅ ∉ ran 𝑓) → X𝑥 ∈ dom 𝑓(𝑓‘𝑥) ≠ ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394  ∀wal 1532   = wceq 1534  ∃wex 1774   ∈ wcel 2099   ≠ wne 2930   ∉ wnel 3036  ∀wral 3051  Vcvv 3462   ∖ cdif 3944  ∅c0 4325  {csn 4633   ↦ cmpt 5236   I cid 5579  dom cdm 5682  ran crn 5683   ↾ cres 5684  Fun wfun 6548   Fn wfn 6549  ‘cfv 6554  Xcixp 8926  CHOICEwac 10158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ixp 8927  df-ac 10159

This theorem is referenced by:  dfac14  23613  dfac21  42727