Theorem dfac2a 8835

Description: Our Axiom of Choice (in the form of ac3 9167) implies the Axiom of Choice (first form) of [Enderton] p. 49. The proof uses neither AC nor the Axiom of Regularity. See dfac2 8836 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 6517	. . . . . . . . 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 6525	. . . . . . . . 9 ⊢ (∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → (℩𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) ∈ 𝑧)
3	1, 2	eqeltrrd 2689	. . . . . . . 8 ⊢ (∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → ∪ {𝑤 ∈ 𝑧 ∣ ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ∈ 𝑧)
4		elequ2 1991	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑧 → (𝑤 ∈ 𝑢 ↔ 𝑤 ∈ 𝑧))
5		elequ1 1984	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑧 → (𝑢 ∈ 𝑣 ↔ 𝑧 ∈ 𝑣))
6	5	anbi1d 737	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑧 → ((𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
7	6	rexbidv 3034	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑧 → (∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
8	4, 7	anbi12d 743	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑧 → ((𝑤 ∈ 𝑢 ∧ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) ↔ (𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))))
9	8	rabbidva2 3162	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑧 → {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} = {𝑤 ∈ 𝑧 ∣ ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)})
10	9	unieqd 4382	. . . . . . . . . 10 ⊢ (𝑢 = 𝑧 → ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} = ∪ {𝑤 ∈ 𝑧 ∣ ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)})
11		eqid 2610	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)}) = (𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)})
12		vex 3176	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
13	12	rabex 4740	. . . . . . . . . . 11 ⊢ {𝑤 ∈ 𝑧 ∣ ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ∈ V
14	13	uniex 6851	. . . . . . . . . 10 ⊢ ∪ {𝑤 ∈ 𝑧 ∣ ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ∈ V
15	10, 11, 14	fvmpt 6191	. . . . . . . . 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 2672	. . . . . . . 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 235	. . . . . . 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 55	. . . . . 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 2934	. . . . 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 3650	. . . . . . . . . . 11 ⊢ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ⊆ 𝑢
21		elssuni 4403	. . . . . . . . . . 11 ⊢ (𝑢 ∈ 𝑥 → 𝑢 ⊆ ∪ 𝑥)
22	20, 21	syl5ss 3579	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑥 → {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ⊆ ∪ 𝑥)
23	22	unissd 4398	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝑥 → ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ⊆ ∪ ∪ 𝑥)
24		vex 3176	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
25	24	uniex 6851	. . . . . . . . . . 11 ⊢ ∪ 𝑥 ∈ V
26	25	uniex 6851	. . . . . . . . . 10 ⊢ ∪ ∪ 𝑥 ∈ V
27	26	elpw2 4755	. . . . . . . . 9 ⊢ (∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ∈ 𝒫 ∪ ∪ 𝑥 ↔ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ⊆ ∪ ∪ 𝑥)
28	23, 27	sylibr 223	. . . . . . . 8 ⊢ (𝑢 ∈ 𝑥 → ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ∈ 𝒫 ∪ ∪ 𝑥)
29	11, 28	fmpti 6291	. . . . . . 7 ⊢ (𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)}):𝑥⟶𝒫 ∪ ∪ 𝑥
30	26	pwex 4774	. . . . . . 7 ⊢ 𝒫 ∪ ∪ 𝑥 ∈ V
31		fex2 7014	. . . . . . 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 1416	. . . . . 6 ⊢ (𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)}) ∈ V
33		fveq1 6102	. . . . . . . . 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 2672	. . . . . . . 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 329	. . . . . . 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 2969	. . . . . 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 3273	. . . . 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 1845	. . 3 ⊢ (∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
40	39	alimi 1730	. 2 ⊢ (∀𝑥∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
41		dfac3 8827	. 2 ⊢ (CHOICE ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
42	40, 41	sylibr 223	1 ⊢ (∀𝑥∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → CHOICE)

Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  ℩crio 6510  CHOICEwac 8821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-riota 6511  df-ac 8822

This theorem is referenced by:  dfac2  8836  axac2  9171