Theorem dfac3 10077

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 10072	. 2 ⊢ (CHOICE ↔ ∀𝑦∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦))
2		vex 3458	. . . . . . . 8 ⊢ 𝑥 ∈ V
3		vuniex 7722	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ V
4	2, 3	xpex 7736	. . . . . . 7 ⊢ (𝑥 × ∪ 𝑥) ∈ V
5		simpl 486	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤) → 𝑤 ∈ 𝑥)
6		elunii 4870	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑣 ∈ ∪ 𝑥)
7	6	ancoms 462	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤) → 𝑣 ∈ ∪ 𝑥)
8	5, 7	jca 519	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤) → (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ ∪ 𝑥))
9	8	ssopab2i 5521	. . . . . . . 8 ⊢ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ ∪ 𝑥)}
10		df-xp 5653	. . . . . . . 8 ⊢ (𝑥 × ∪ 𝑥) = {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ ∪ 𝑥)}
11	9, 10	sseqtrri 3985	. . . . . . 7 ⊢ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ⊆ (𝑥 × ∪ 𝑥)
12	4, 11	ssexi 5278	. . . . . 6 ⊢ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ∈ V
13		sseq2 3962	. . . . . . . 8 ⊢ (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} → (𝑓 ⊆ 𝑦 ↔ 𝑓 ⊆ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}))
14		dmeq 5879	. . . . . . . . 9 ⊢ (𝑦 = {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} → dom 𝑦 = dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)})
15	14	fneq2d 6615	. . . . . . . 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 641	. . . . . . 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 1941	. . . . . 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 3564	. . . . 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 6624	. . . . . . . . . . . . 13 ⊢ (𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} → dom 𝑓 = dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)})
20		dmopab 5891	. . . . . . . . . . . . . . . 16 ⊢ dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} = {𝑤 ∣ ∃𝑣(𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}
21	20	eleq2i 2854	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ↔ 𝑧 ∈ {𝑤 ∣ ∃𝑣(𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)})
22		vex 3458	. . . . . . . . . . . . . . . 16 ⊢ 𝑧 ∈ V
23		elequ1 2149	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥))
24		eleq2 2851	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑧 → (𝑣 ∈ 𝑤 ↔ 𝑣 ∈ 𝑧))
25	23, 24	anbi12d 641	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑧 → ((𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤) ↔ (𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧)))
26	25	exbidv 1941	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑧 → (∃𝑣(𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤) ↔ ∃𝑣(𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧)))
27	22, 26	elab 3638	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ {𝑤 ∣ ∃𝑣(𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} ↔ ∃𝑣(𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧))
28		19.42v 1973	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑣(𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧) ↔ (𝑧 ∈ 𝑥 ∧ ∃𝑣 𝑣 ∈ 𝑧))
29		n0 4305	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ≠ ∅ ↔ ∃𝑣 𝑣 ∈ 𝑧)
30	29	anbi2i 632	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) ↔ (𝑧 ∈ 𝑥 ∧ ∃𝑣 𝑣 ∈ 𝑧))
31	28, 30	bitr4i 280	. . . . . . . . . . . . . . 15 ⊢ (∃𝑣(𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧) ↔ (𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅))
32	21, 27, 31	3bitrri 300	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅) ↔ 𝑧 ∈ dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)})
33		eleq2 2851	. . . . . . . . . . . . . 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 292	. . . . . . . . . . . . 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 485	. . . . . . . . . . 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 6621	. . . . . . . . . . . 12 ⊢ (𝑓 Fn dom {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} → Fun 𝑓)
38		funfvima3 7220	. . . . . . . . . . . . 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 462	. . . . . . . . . . . 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 602	. . . . . . . . . . 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 242	. . . . . . . . . 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 410	. . . . . . . . 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 6073	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ V → ({⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} “ {𝑧}) = {𝑢 ∣ 𝑧{⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}𝑢})
44	43	elv 3459	. . . . . . . . . . . . 13 ⊢ ({⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} “ {𝑧}) = {𝑢 ∣ 𝑧{⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}𝑢}
45		vex 3458	. . . . . . . . . . . . . . 15 ⊢ 𝑢 ∈ V
46		elequ1 2149	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑢 → (𝑣 ∈ 𝑧 ↔ 𝑢 ∈ 𝑧))
47	46	anbi2d 639	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑢 → ((𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧) ↔ (𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧)))
48		eqid 2762	. . . . . . . . . . . . . . 15 ⊢ {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} = {⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}
49	22, 45, 25, 47, 48	brab 5514	. . . . . . . . . . . . . 14 ⊢ (𝑧{⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}𝑢 ↔ (𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧))
50	49	abbii 2829	. . . . . . . . . . . . 13 ⊢ {𝑢 ∣ 𝑧{⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)}𝑢} = {𝑢 ∣ (𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧)}
51	44, 50	eqtri 2785	. . . . . . . . . . . 12 ⊢ ({⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} “ {𝑧}) = {𝑢 ∣ (𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧)}
52		ibar 536	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑥 → (𝑢 ∈ 𝑧 ↔ (𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧)))
53	52	eqabdv 2895	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑥 → 𝑧 = {𝑢 ∣ (𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧)})
54	51, 53	eqtr4id 2816	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑥 → ({⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} “ {𝑧}) = 𝑧)
55	54	eleq2d 2848	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑥 → ((𝑓‘𝑧) ∈ ({⟨𝑤, 𝑣⟩ ∣ (𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤)} “ {𝑧}) ↔ (𝑓‘𝑧) ∈ 𝑧))
56	55	ad2antrl 738	. . . . . . . . 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 234	. . . . . . . 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 424	. . . . . . 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 3153	. . . . . 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 1855	. . . . 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 1947	. . 3 ⊢ (∀𝑦∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦) → ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
63		eqid 2762	. . . . 5 ⊢ (𝑤 ∈ dom 𝑦 ↦ (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢})) = (𝑤 ∈ dom 𝑦 ↦ (𝑓‘{𝑢 ∣ 𝑤𝑦𝑢}))
64	63	aceq3lem 10076	. . . 4 ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦))
65	64	alrimiv 1947	. . 3 ⊢ (∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) → ∀𝑦∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦))
66	62, 65	impbii 211	. 2 ⊢ (∀𝑦∃𝑓(𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦) ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
67	1, 66	bitri 277	1 ⊢ (CHOICE ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1558   = wceq 1560  ∃wex 1799   ∈ wcel 2142  {cab 2740   ≠ wne 2957  ∀wral 3076  Vcvv 3454   ⊆ wss 3904  ∅c0 4285  {csn 4582  ∪ cuni 4865   class class class wbr 5100  {copab 5162   ↦ cmpt 5181   × cxp 5645  dom cdm 5647   “ cima 5650  Fun wfun 6515   Fn wfn 6516  ‘cfv 6521  CHOICEwac 10071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-fv 6529  df-ac 10072

This theorem is referenced by:  dfac4  10078  dfac5  10085  dfac2a  10086  dfac2b  10087  dfac8  10092  dfac9  10093  ac4  10432  dfac11  43639