Theorem dfac3 10200

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 10195	. 2 ⊢ (CHOICE ↔ ∀𝑦∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦))
2		vex 3455	. . . . . . . 8 ⊢ 𝑥 ∈ V
3		vuniex 7756	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ V
4	2, 3	xpex 7767	. . . . . . 7 ⊢ (𝑥 × ∪ 𝑥) ∈ V
5		simpl 488	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤) → 𝑤 ∈ 𝑥)
6		elunii 4872	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑣 ∈ ∪ 𝑥)
7	6	ancoms 464	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤) → 𝑣 ∈ ∪ 𝑥)
8	5, 7	jca 521	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤) → (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ ∪ 𝑥))
9	8	ssopab2i 5525	. . . . . . . 8 ⊢ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ ∪ 𝑥)}
10		df-xp 5657	. . . . . . . 8 ⊢ (𝑥 × ∪ 𝑥) = {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ ∪ 𝑥)}
11	9, 10	sseqtrri 3980	. . . . . . 7 ⊢ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ⊆ (𝑥 × ∪ 𝑥)
12	4, 11	ssexi 5284	. . . . . 6 ⊢ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ∈ V
13		sseq2 3957	. . . . . . . 8 ⊢ (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} → (𝑓 ⊆ 𝑦 ↔ 𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}))
14		dmeq 5885	. . . . . . . . 9 ⊢ (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} → dom 𝑦 = dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)})
15	14	fneq2d 6633	. . . . . . . 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 644	. . . . . . 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 1954	. . . . . 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 3560	. . . . 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 6642	. . . . . . . . . . . . 13 ⊢ (𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} → dom 𝑓 = dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)})
20		dmopab 5897	. . . . . . . . . . . . . . . 16 ⊢ dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} = {𝑤 ∣ ∃𝑣(𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}
21	20	eleq2i 2853	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ↔ 𝑧 ∈ {𝑤 ∣ ∃𝑣(𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)})
22		vex 3455	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ V
23		elequ1 2152	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥))
24		eleq2 2850	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → (𝑣 ∈ 𝑤 ↔ 𝑣 ∈ 𝑧))
25	23, 24	anbi12d 644	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑧 → ((𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤) ↔ (𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧)))
26	25	exbidv 1954	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑧 → (∃𝑣(𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤) ↔ ∃𝑣(𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧)))
27	22, 26	elab 3633	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ {𝑤 ∣ ∃𝑣(𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ↔ ∃𝑣(𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧))
28		19.42v 1986	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑣(𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧) ↔ (𝑧 ∈ 𝑥 ∧ ∃𝑣 𝑣 ∈ 𝑧))
29		n0 4300	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ≠ ∅ ↔ ∃𝑣 𝑣 ∈ 𝑧)
30	29	anbi2i 635	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) ↔ (𝑧 ∈ 𝑥 ∧ ∃𝑣 𝑣 ∈ 𝑧))
31	28, 30	bitr4i 281	. . . . . . . . . . . . . . 15 ⊢ (∃𝑣(𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧) ↔ (𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅))
32	21, 27, 31	3bitrri 301	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) ↔ 𝑧 ∈ dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)})
33		eleq2 2850	. . . . . . . . . . . . . 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 293	. . . . . . . . . . . . 13 ⊢ (dom 𝑓 = dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} → ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) ↔ 𝑧 ∈ dom 𝑓))
35	19, 34	syl 18	. . . . . . . . . . . 12 ⊢ (𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} → ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) ↔ 𝑧 ∈ dom 𝑓))
36	35	adantl 487	. . . . . . . . . . 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 6639	. . . . . . . . . . . 12 ⊢ (𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} → Fun 𝑓)
38		funfvima3 7242	. . . . . . . . . . . . 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 464	. . . . . . . . . . . 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 605	. . . . . . . . . . 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 243	. . . . . . . . . 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 412	. . . . . . . . 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 6082	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ V → ({⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} “ {𝑧}) = {𝑢 ∣ 𝑧{⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}𝑢})
44	43	elv 3456	. . . . . . . . . . . . 13 ⊢ ({⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} “ {𝑧}) = {𝑢 ∣ 𝑧{⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}𝑢}
45		vex 3455	. . . . . . . . . . . . . . 15 ⊢ 𝑢 ∈ V
46		elequ1 2152	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑢 → (𝑣 ∈ 𝑧 ↔ 𝑢 ∈ 𝑧))
47	46	anbi2d 642	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑢 → ((𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧) ↔ (𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧)))
48		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} = {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}
49	22, 45, 25, 47, 48	brab 5518	. . . . . . . . . . . . . 14 ⊢ (𝑧{⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}𝑢 ↔ (𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧))
50	49	abbii 2828	. . . . . . . . . . . . 13 ⊢ {𝑢 ∣ 𝑧{⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}𝑢} = {𝑢 ∣ (𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧)}
51	44, 50	eqtri 2784	. . . . . . . . . . . 12 ⊢ ({⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} “ {𝑧}) = {𝑢 ∣ (𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧)}
52		ibar 538	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑥 → (𝑢 ∈ 𝑧 ↔ (𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧)))
53	52	eqabdv 2894	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑥 → 𝑧 = {𝑢 ∣ (𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧)})
54	51, 53	eqtr4id 2815	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑥 → ({⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} “ {𝑧}) = 𝑧)
55	54	eleq2d 2847	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑥 → ((𝑓‘𝑧) ∈ ({⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} “ {𝑧}) ↔ (𝑓‘𝑧) ∈ 𝑧))
56	55	ad2antrl 741	. . . . . . . . 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 235	. . . . . . . 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 426	. . . . . . 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 3154	. . . . . 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 1868	. . . . 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 18	. . . 4 ⊢ (∀𝑦∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦) → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
62	61	alrimiv 1960	. . 3 ⊢ (∀𝑦∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦) → ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
63		eqid 2761	. . . . 5 ⊢ (𝑤 ∈ dom 𝑦 ↦ (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢})) = (𝑤 ∈ dom 𝑦 ↦ (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}))
64	63	aceq3lem 10199	. . . 4 ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦))
65	64	alrimiv 1960	. . 3 ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∀𝑦∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦))
66	62, 65	impbii 212	. 2 ⊢ (∀𝑦∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦) ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
67	1, 66	bitri 278	1 ⊢ (CHOICE ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2739   ≠ wne 2956  ∀wral 3077  Vcvv 3451   ⊆ wss 3899  ∅c0 4279  {csn 4584  ∪ cuni 4867   class class class wbr 5103  {copab 5167   ↦ cmpt 5186   × cxp 5649  dom cdm 5651   “ cima 5654  Fun wfun 6532   Fn wfn 6533  ‘cfv 6538  CHOICEwac 10194

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546  df-ac 10195

This theorem is used by:  dfac4  10201  dfac5  10207  dfac2a  10208  dfac2b  10209  dfac8  10214  dfac9  10215  ac4  10553  dfac11  44063