Theorem dfac3 10159

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 10154	. 2 ⊢ (CHOICE ↔ ∀𝑦∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦))
2		vex 3482	. . . . . . . 8 ⊢ 𝑥 ∈ V
3		vuniex 7758	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ V
4	2, 3	xpex 7772	. . . . . . 7 ⊢ (𝑥 × ∪ 𝑥) ∈ V
5		simpl 482	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤) → 𝑤 ∈ 𝑥)
6		elunii 4917	. . . . . . . . . . 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 5560	. . . . . . . 8 ⊢ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ ∪ 𝑥)}
10		df-xp 5695	. . . . . . . 8 ⊢ (𝑥 × ∪ 𝑥) = {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ ∪ 𝑥)}
11	9, 10	sseqtrri 4033	. . . . . . 7 ⊢ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ⊆ (𝑥 × ∪ 𝑥)
12	4, 11	ssexi 5328	. . . . . 6 ⊢ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ∈ V
13		sseq2 4022	. . . . . . . 8 ⊢ (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} → (𝑓 ⊆ 𝑦 ↔ 𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}))
14		dmeq 5917	. . . . . . . . 9 ⊢ (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} → dom 𝑦 = dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)})
15	14	fneq2d 6663	. . . . . . . 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 632	. . . . . . 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 1919	. . . . . 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 3605	. . . . 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 6672	. . . . . . . . . . . . 13 ⊢ (𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} → dom 𝑓 = dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)})
20		dmopab 5929	. . . . . . . . . . . . . . . 16 ⊢ dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} = {𝑤 ∣ ∃𝑣(𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}
21	20	eleq2i 2831	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ↔ 𝑧 ∈ {𝑤 ∣ ∃𝑣(𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)})
22		vex 3482	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ V
23		elequ1 2113	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥))
24		eleq2 2828	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → (𝑣 ∈ 𝑤 ↔ 𝑣 ∈ 𝑧))
25	23, 24	anbi12d 632	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑧 → ((𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤) ↔ (𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧)))
26	25	exbidv 1919	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑧 → (∃𝑣(𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤) ↔ ∃𝑣(𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧)))
27	22, 26	elab 3681	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ {𝑤 ∣ ∃𝑣(𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ↔ ∃𝑣(𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧))
28		19.42v 1951	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑣(𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧) ↔ (𝑧 ∈ 𝑥 ∧ ∃𝑣 𝑣 ∈ 𝑧))
29		n0 4359	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ≠ ∅ ↔ ∃𝑣 𝑣 ∈ 𝑧)
30	29	anbi2i 623	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) ↔ (𝑧 ∈ 𝑥 ∧ ∃𝑣 𝑣 ∈ 𝑧))
31	28, 30	bitr4i 278	. . . . . . . . . . . . . . 15 ⊢ (∃𝑣(𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧) ↔ (𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅))
32	21, 27, 31	3bitrri 298	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) ↔ 𝑧 ∈ dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)})
33		eleq2 2828	. . . . . . . . . . . . . 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 290	. . . . . . . . . . . . 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 6669	. . . . . . . . . . . 12 ⊢ (𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} → Fun 𝑓)
38		funfvima3 7256	. . . . . . . . . . . . 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 593	. . . . . . . . . . 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 240	. . . . . . . . . 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 6104	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ V → ({⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} “ {𝑧}) = {𝑢 ∣ 𝑧{⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}𝑢})
44	43	elv 3483	. . . . . . . . . . . . 13 ⊢ ({⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} “ {𝑧}) = {𝑢 ∣ 𝑧{⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}𝑢}
45		vex 3482	. . . . . . . . . . . . . . 15 ⊢ 𝑢 ∈ V
46		elequ1 2113	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑢 → (𝑣 ∈ 𝑧 ↔ 𝑢 ∈ 𝑧))
47	46	anbi2d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑢 → ((𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧) ↔ (𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧)))
48		eqid 2735	. . . . . . . . . . . . . . 15 ⊢ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} = {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}
49	22, 45, 25, 47, 48	brab 5553	. . . . . . . . . . . . . 14 ⊢ (𝑧{⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}𝑢 ↔ (𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧))
50	49	abbii 2807	. . . . . . . . . . . . 13 ⊢ {𝑢 ∣ 𝑧{⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}𝑢} = {𝑢 ∣ (𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧)}
51	44, 50	eqtri 2763	. . . . . . . . . . . 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	eqabdv 2873	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑥 → 𝑧 = {𝑢 ∣ (𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧)})
54	51, 53	eqtr4id 2794	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑥 → ({⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} “ {𝑧}) = 𝑧)
55	54	eleq2d 2825	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑥 → ((𝑓‘𝑧) ∈ ({⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} “ {𝑧}) ↔ (𝑓‘𝑧) ∈ 𝑧))
56	55	ad2antrl 728	. . . . . . . . 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 232	. . . . . . . 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 3143	. . . . . 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 1832	. . . . 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 1925	. . 3 ⊢ (∀𝑦∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦) → ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
63		eqid 2735	. . . . 5 ⊢ (𝑤 ∈ dom 𝑦 ↦ (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢})) = (𝑤 ∈ dom 𝑦 ↦ (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}))
64	63	aceq3lem 10158	. . . 4 ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦))
65	64	alrimiv 1925	. . 3 ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∀𝑦∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦))
66	62, 65	impbii 209	. 2 ⊢ (∀𝑦∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦) ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
67	1, 66	bitri 275	1 ⊢ (CHOICE ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535   = wceq 1537  ∃wex 1776   ∈ wcel 2106  {cab 2712   ≠ wne 2938  ∀wral 3059  Vcvv 3478   ⊆ wss 3963  ∅c0 4339  {csn 4631  ∪ cuni 4912   class class class wbr 5148  {copab 5210   ↦ cmpt 5231   × cxp 5687  dom cdm 5689   “ cima 5692  Fun wfun 6557   Fn wfn 6558  ‘cfv 6563  CHOICEwac 10153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-ac 10154

This theorem is referenced by:  dfac4  10160  dfac5  10167  dfac2a  10168  dfac2b  10169  dfac8  10174  dfac9  10175  ac4  10513  dfac11  43051