Theorem dfac2a 8952

Description: Our Axiom of Choice (in the form of ac3 9284) implies the Axiom of Choice (first form) of [Enderton] p. 49. The proof uses neither AC nor the Axiom of Regularity. See dfac2 8953 for the converse (which does use the Axiom of Regularity). (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)

Assertion

Ref	Expression
dfac2a	⊢ (∀𝑥∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → CHOICE)

Distinct variable group:   𝑥,𝑧,𝑦,𝑤,𝑣

Proof of Theorem dfac2a

Dummy variables 𝑓 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		riotauni 6617	. . . . . . . . 9 ⊢ (∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → (℩𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) = ∪ {𝑤 ∈ 𝑧 ∣ ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)})
2		riotacl 6625	. . . . . . . . 9 ⊢ (∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → (℩𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) ∈ 𝑧)
3	1, 2	eqeltrrd 2702	. . . . . . . 8 ⊢ (∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → ∪ {𝑤 ∈ 𝑧 ∣ ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ∈ 𝑧)
4		elequ2 2004	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑧 → (𝑤 ∈ 𝑢 ↔ 𝑤 ∈ 𝑧))
5		elequ1 1997	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑧 → (𝑢 ∈ 𝑣 ↔ 𝑧 ∈ 𝑣))
6	5	anbi1d 741	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑧 → ((𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
7	6	rexbidv 3052	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑧 → (∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
8	4, 7	anbi12d 747	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑧 → ((𝑤 ∈ 𝑢 ∧ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) ↔ (𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))))
9	8	rabbidva2 3186	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑧 → {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} = {𝑤 ∈ 𝑧 ∣ ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)})
10	9	unieqd 4446	. . . . . . . . . 10 ⊢ (𝑢 = 𝑧 → ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} = ∪ {𝑤 ∈ 𝑧 ∣ ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)})
11		eqid 2622	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)}) = (𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)})
12		vex 3203	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
13	12	rabex 4813	. . . . . . . . . . 11 ⊢ {𝑤 ∈ 𝑧 ∣ ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ∈ V
14	13	uniex 6953	. . . . . . . . . 10 ⊢ ∪ {𝑤 ∈ 𝑧 ∣ ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ∈ V
15	10, 11, 14	fvmpt 6282	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑥 → ((𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)})‘𝑧) = ∪ {𝑤 ∈ 𝑧 ∣ ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)})
16	15	eleq1d 2686	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑥 → (((𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)})‘𝑧) ∈ 𝑧 ↔ ∪ {𝑤 ∈ 𝑧 ∣ ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ∈ 𝑧))
17	3, 16	syl5ibr 236	. . . . . . 7 ⊢ (𝑧 ∈ 𝑥 → (∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → ((𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)})‘𝑧) ∈ 𝑧))
18	17	imim2d 57	. . . . . 6 ⊢ (𝑧 ∈ 𝑥 → ((𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → (𝑧 ≠ ∅ → ((𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)})‘𝑧) ∈ 𝑧)))
19	18	ralimia 2950	. . . . 5 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)})‘𝑧) ∈ 𝑧))
20		ssrab2 3687	. . . . . . . . . . 11 ⊢ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ⊆ 𝑢
21		elssuni 4467	. . . . . . . . . . 11 ⊢ (𝑢 ∈ 𝑥 → 𝑢 ⊆ ∪ 𝑥)
22	20, 21	syl5ss 3614	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑥 → {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ⊆ ∪ 𝑥)
23	22	unissd 4462	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝑥 → ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ⊆ ∪ ∪ 𝑥)
24		vex 3203	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
25	24	uniex 6953	. . . . . . . . . . 11 ⊢ ∪ 𝑥 ∈ V
26	25	uniex 6953	. . . . . . . . . 10 ⊢ ∪ ∪ 𝑥 ∈ V
27	26	elpw2 4828	. . . . . . . . 9 ⊢ (∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ∈ 𝒫 ∪ ∪ 𝑥 ↔ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ⊆ ∪ ∪ 𝑥)
28	23, 27	sylibr 224	. . . . . . . 8 ⊢ (𝑢 ∈ 𝑥 → ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ∈ 𝒫 ∪ ∪ 𝑥)
29	11, 28	fmpti 6383	. . . . . . 7 ⊢ (𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)}):𝑥⟶𝒫 ∪ ∪ 𝑥
30	26	pwex 4848	. . . . . . 7 ⊢ 𝒫 ∪ ∪ 𝑥 ∈ V
31		fex2 7121	. . . . . . 7 ⊢ (((𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)}):𝑥⟶𝒫 ∪ ∪ 𝑥 ∧ 𝑥 ∈ V ∧ 𝒫 ∪ ∪ 𝑥 ∈ V) → (𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)}) ∈ V)
32	29, 24, 30, 31	mp3an 1424	. . . . . 6 ⊢ (𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)}) ∈ V
33		fveq1 6190	. . . . . . . . 9 ⊢ (𝑓 = (𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)}) → (𝑓‘𝑧) = ((𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)})‘𝑧))
34	33	eleq1d 2686	. . . . . . . 8 ⊢ (𝑓 = (𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)}) → ((𝑓‘𝑧) ∈ 𝑧 ↔ ((𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)})‘𝑧) ∈ 𝑧))
35	34	imbi2d 330	. . . . . . 7 ⊢ (𝑓 = (𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)}) → ((𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → ((𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)})‘𝑧) ∈ 𝑧)))
36	35	ralbidv 2986	. . . . . 6 ⊢ (𝑓 = (𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)}) → (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)})‘𝑧) ∈ 𝑧)))
37	32, 36	spcev 3300	. . . . 5 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ((𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)})‘𝑧) ∈ 𝑧) → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
38	19, 37	syl 17	. . . 4 ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
39	38	exlimiv 1858	. . 3 ⊢ (∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
40	39	alimi 1739	. 2 ⊢ (∀𝑥∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
41		dfac3 8944	. 2 ⊢ (CHOICE ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
42	40, 41	sylibr 224	1 ⊢ (∀𝑥∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → CHOICE)

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  ∃!wreu 2914  {crab 2916  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  ∪ cuni 4436   ↦ cmpt 4729  ⟶wf 5884  ‘cfv 5888  ℩crio 6610  CHOICEwac 8938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-riota 6611  df-ac 8939

This theorem is referenced by:  dfac2  8953  axac2  9288