Theorem dfac3 9808

Description: Equivalence of two versions of the Axiom of Choice. The left-hand side is defined as the Axiom of Choice (first form) of [Enderton] p. 49. The right-hand side is the Axiom of Choice of [TakeutiZaring] p. 83. The proof does not depend on AC. (Contributed by NM, 24-Mar-2004.) (Revised by Stefan O'Rear, 22-Feb-2015.)

Assertion

Ref	Expression
dfac3	⊢ (CHOICE ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))

Distinct variable group:   𝑥,𝑓,𝑧

Proof of Theorem dfac3

Dummy variables 𝑦 𝑤 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ac 9803	. 2 ⊢ (CHOICE ↔ ∀𝑦∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦))
2		vex 3426	. . . . . . . 8 ⊢ 𝑥 ∈ V
3		vuniex 7570	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ V
4	2, 3	xpex 7581	. . . . . . 7 ⊢ (𝑥 × ∪ 𝑥) ∈ V
5		simpl 482	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤) → 𝑤 ∈ 𝑥)
6		elunii 4841	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑣 ∈ ∪ 𝑥)
7	6	ancoms 458	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤) → 𝑣 ∈ ∪ 𝑥)
8	5, 7	jca 511	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤) → (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ ∪ 𝑥))
9	8	ssopab2i 5456	. . . . . . . 8 ⊢ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ ∪ 𝑥)}
10		df-xp 5586	. . . . . . . 8 ⊢ (𝑥 × ∪ 𝑥) = {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ ∪ 𝑥)}
11	9, 10	sseqtrri 3954	. . . . . . 7 ⊢ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ⊆ (𝑥 × ∪ 𝑥)
12	4, 11	ssexi 5241	. . . . . 6 ⊢ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ∈ V
13		sseq2 3943	. . . . . . . 8 ⊢ (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} → (𝑓 ⊆ 𝑦 ↔ 𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}))
14		dmeq 5801	. . . . . . . . 9 ⊢ (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} → dom 𝑦 = dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)})
15	14	fneq2d 6511	. . . . . . . 8 ⊢ (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} → (𝑓 Fn dom 𝑦 ↔ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}))
16	13, 15	anbi12d 630	. . . . . . 7 ⊢ (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} → ((𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦) ↔ (𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)})))
17	16	exbidv 1925	. . . . . 6 ⊢ (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} → (∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦) ↔ ∃𝑓(𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)})))
18	12, 17	spcv 3534	. . . . 5 ⊢ (∀𝑦∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦) → ∃𝑓(𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}))
19		fndm 6520	. . . . . . . . . . . . 13 ⊢ (𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} → dom 𝑓 = dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)})
20		dmopab 5813	. . . . . . . . . . . . . . . 16 ⊢ dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} = {𝑤 ∣ ∃𝑣(𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}
21	20	eleq2i 2830	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ↔ 𝑧 ∈ {𝑤 ∣ ∃𝑣(𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)})
22		vex 3426	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ V
23		elequ1 2115	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥))
24		eleq2 2827	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → (𝑣 ∈ 𝑤 ↔ 𝑣 ∈ 𝑧))
25	23, 24	anbi12d 630	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑧 → ((𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤) ↔ (𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧)))
26	25	exbidv 1925	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑧 → (∃𝑣(𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤) ↔ ∃𝑣(𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧)))
27	22, 26	elab 3602	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ {𝑤 ∣ ∃𝑣(𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ↔ ∃𝑣(𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧))
28		19.42v 1958	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑣(𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧) ↔ (𝑧 ∈ 𝑥 ∧ ∃𝑣 𝑣 ∈ 𝑧))
29		n0 4277	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ≠ ∅ ↔ ∃𝑣 𝑣 ∈ 𝑧)
30	29	anbi2i 622	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) ↔ (𝑧 ∈ 𝑥 ∧ ∃𝑣 𝑣 ∈ 𝑧))
31	28, 30	bitr4i 277	. . . . . . . . . . . . . . 15 ⊢ (∃𝑣(𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧) ↔ (𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅))
32	21, 27, 31	3bitrri 297	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) ↔ 𝑧 ∈ dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)})
33		eleq2 2827	. . . . . . . . . . . . . 14 ⊢ (dom 𝑓 = dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} → (𝑧 ∈ dom 𝑓 ↔ 𝑧 ∈ dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}))
34	32, 33	bitr4id 289	. . . . . . . . . . . . 13 ⊢ (dom 𝑓 = dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} → ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) ↔ 𝑧 ∈ dom 𝑓))
35	19, 34	syl 17	. . . . . . . . . . . 12 ⊢ (𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} → ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) ↔ 𝑧 ∈ dom 𝑓))
36	35	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}) → ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) ↔ 𝑧 ∈ dom 𝑓))
37		fnfun 6517	. . . . . . . . . . . 12 ⊢ (𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} → Fun 𝑓)
38		funfvima3 7094	. . . . . . . . . . . . 13 ⊢ ((Fun 𝑓 ∧ 𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}) → (𝑧 ∈ dom 𝑓 → (𝑓‘𝑧) ∈ ({⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} “ {𝑧})))
39	38	ancoms 458	. . . . . . . . . . . 12 ⊢ ((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ∧ Fun 𝑓) → (𝑧 ∈ dom 𝑓 → (𝑓‘𝑧) ∈ ({⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} “ {𝑧})))
40	37, 39	sylan2 592	. . . . . . . . . . 11 ⊢ ((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}) → (𝑧 ∈ dom 𝑓 → (𝑓‘𝑧) ∈ ({⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} “ {𝑧})))
41	36, 40	sylbid 239	. . . . . . . . . 10 ⊢ ((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}) → ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) → (𝑓‘𝑧) ∈ ({⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} “ {𝑧})))
42	41	imp 406	. . . . . . . . 9 ⊢ (((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}) ∧ (𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅)) → (𝑓‘𝑧) ∈ ({⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} “ {𝑧}))
43		imasng 5980	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ V → ({⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} “ {𝑧}) = {𝑢 ∣ 𝑧{⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}𝑢})
44	43	elv 3428	. . . . . . . . . . . . 13 ⊢ ({⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} “ {𝑧}) = {𝑢 ∣ 𝑧{⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}𝑢}
45		vex 3426	. . . . . . . . . . . . . . 15 ⊢ 𝑢 ∈ V
46		elequ1 2115	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑢 → (𝑣 ∈ 𝑧 ↔ 𝑢 ∈ 𝑧))
47	46	anbi2d 628	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑢 → ((𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧) ↔ (𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧)))
48		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} = {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}
49	22, 45, 25, 47, 48	brab 5449	. . . . . . . . . . . . . 14 ⊢ (𝑧{⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}𝑢 ↔ (𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧))
50	49	abbii 2809	. . . . . . . . . . . . 13 ⊢ {𝑢 ∣ 𝑧{⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}𝑢} = {𝑢 ∣ (𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧)}
51	44, 50	eqtri 2766	. . . . . . . . . . . 12 ⊢ ({⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} “ {𝑧}) = {𝑢 ∣ (𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧)}
52		ibar 528	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑥 → (𝑢 ∈ 𝑧 ↔ (𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧)))
53	52	abbi2dv 2876	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑥 → 𝑧 = {𝑢 ∣ (𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧)})
54	51, 53	eqtr4id 2798	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑥 → ({⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} “ {𝑧}) = 𝑧)
55	54	eleq2d 2824	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑥 → ((𝑓‘𝑧) ∈ ({⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} “ {𝑧}) ↔ (𝑓‘𝑧) ∈ 𝑧))
56	55	ad2antrl 724	. . . . . . . . 9 ⊢ (((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}) ∧ (𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅)) → ((𝑓‘𝑧) ∈ ({⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} “ {𝑧}) ↔ (𝑓‘𝑧) ∈ 𝑧))
57	42, 56	mpbid 231	. . . . . . . 8 ⊢ (((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}) ∧ (𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅)) → (𝑓‘𝑧) ∈ 𝑧)
58	57	exp32 420	. . . . . . 7 ⊢ ((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}) → (𝑧 ∈ 𝑥 → (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
59	58	ralrimiv 3106	. . . . . 6 ⊢ ((𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}) → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
60	59	eximi 1838	. . . . 5 ⊢ (∃𝑓(𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ∧ 𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}) → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
61	18, 60	syl 17	. . . 4 ⊢ (∀𝑦∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦) → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
62	61	alrimiv 1931	. . 3 ⊢ (∀𝑦∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦) → ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
63		eqid 2738	. . . . 5 ⊢ (𝑤 ∈ dom 𝑦 ↦ (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢})) = (𝑤 ∈ dom 𝑦 ↦ (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}))
64	63	aceq3lem 9807	. . . 4 ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦))
65	64	alrimiv 1931	. . 3 ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∀𝑦∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦))
66	62, 65	impbii 208	. 2 ⊢ (∀𝑦∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦) ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
67	1, 66	bitri 274	1 ⊢ (CHOICE ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715   ≠ wne 2942  ∀wral 3063  Vcvv 3422   ⊆ wss 3883  ∅c0 4253  {csn 4558  ∪ cuni 4836   class class class wbr 5070  {copab 5132   ↦ cmpt 5153   × cxp 5578  dom cdm 5580   “ cima 5583  Fun wfun 6412   Fn wfn 6413  ‘cfv 6418  CHOICEwac 9802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426  df-ac 9803

This theorem is referenced by:  dfac4  9809  dfac5  9815  dfac2a  9816  dfac2b  9817  dfac8  9822  dfac9  9823  ac4  10162  dfac11  40803