Theorem dfac2a 9547

Description: Our Axiom of Choice (in the form of ac3 9876) implies the Axiom of Choice (first form) of [Enderton] p. 49. The proof uses neither AC nor the Axiom of Regularity. See dfac2b 9548 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 7112	. . . . . . . . 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 7123	. . . . . . . . 9 ⊢ (∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → (℩𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) ∈ 𝑧)
3	1, 2	eqeltrrd 2912	. . . . . . . 8 ⊢ (∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) → ∪ {𝑤 ∈ 𝑧 ∣ ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ∈ 𝑧)
4		elequ2 2123	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑧 → (𝑤 ∈ 𝑢 ↔ 𝑤 ∈ 𝑧))
5		elequ1 2115	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑧 → (𝑢 ∈ 𝑣 ↔ 𝑧 ∈ 𝑣))
6	5	anbi1d 631	. . . . . . . . . . . . . 14 ⊢ (𝑢 = 𝑧 → ((𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
7	6	rexbidv 3295	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑧 → (∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣) ↔ ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)))
8	4, 7	anbi12d 632	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑧 → ((𝑤 ∈ 𝑢 ∧ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) ↔ (𝑤 ∈ 𝑧 ∧ ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣))))
9	8	rabbidva2 3475	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑧 → {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} = {𝑤 ∈ 𝑧 ∣ ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)})
10	9	unieqd 4840	. . . . . . . . . 10 ⊢ (𝑢 = 𝑧 → ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} = ∪ {𝑤 ∈ 𝑧 ∣ ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)})
11		eqid 2819	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)}) = (𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)})
12		vex 3496	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
13	12	rabex 5226	. . . . . . . . . . 11 ⊢ {𝑤 ∈ 𝑧 ∣ ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ∈ V
14	13	uniex 7459	. . . . . . . . . 10 ⊢ ∪ {𝑤 ∈ 𝑧 ∣ ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ∈ V
15	10, 11, 14	fvmpt 6761	. . . . . . . . 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 2895	. . . . . . . 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 248	. . . . . . 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 3156	. . . . 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 4054	. . . . . . . . . . 11 ⊢ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ⊆ 𝑢
21		elssuni 4859	. . . . . . . . . . 11 ⊢ (𝑢 ∈ 𝑥 → 𝑢 ⊆ ∪ 𝑥)
22	20, 21	sstrid 3976	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝑥 → {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ⊆ ∪ 𝑥)
23	22	unissd 4854	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝑥 → ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ⊆ ∪ ∪ 𝑥)
24		vex 3496	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
25	24	uniex 7459	. . . . . . . . . . 11 ⊢ ∪ 𝑥 ∈ V
26	25	uniex 7459	. . . . . . . . . 10 ⊢ ∪ ∪ 𝑥 ∈ V
27	26	elpw2 5239	. . . . . . . . 9 ⊢ (∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ∈ 𝒫 ∪ ∪ 𝑥 ↔ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ⊆ ∪ ∪ 𝑥)
28	23, 27	sylibr 236	. . . . . . . 8 ⊢ (𝑢 ∈ 𝑥 → ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)} ∈ 𝒫 ∪ ∪ 𝑥)
29	11, 28	fmpti 6869	. . . . . . 7 ⊢ (𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)}):𝑥⟶𝒫 ∪ ∪ 𝑥
30	26	pwex 5272	. . . . . . 7 ⊢ 𝒫 ∪ ∪ 𝑥 ∈ V
31		fex2 7630	. . . . . . 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 1455	. . . . . 6 ⊢ (𝑢 ∈ 𝑥 ↦ ∪ {𝑤 ∈ 𝑢 ∣ ∃𝑣 ∈ 𝑦 (𝑢 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)}) ∈ V
33		fveq1 6662	. . . . . . . . 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 2895	. . . . . . . 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 343	. . . . . . 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 3195	. . . . . 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 3605	. . . . 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 1925	. . 3 ⊢ (∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → ∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
40	39	alimi 1806	. 2 ⊢ (∀𝑥∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
41		dfac3 9539	. 2 ⊢ (CHOICE ↔ ∀𝑥∃𝑓∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))
42	40, 41	sylibr 236	1 ⊢ (∀𝑥∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑤 ∈ 𝑧 ∃𝑣 ∈ 𝑦 (𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣)) → CHOICE)

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1529   = wceq 1531  ∃wex 1774   ∈ wcel 2108   ≠ wne 3014  ∀wral 3136  ∃wrex 3137  ∃!wreu 3138  {crab 3140  Vcvv 3493   ⊆ wss 3934  ∅c0 4289  𝒫 cpw 4537  ∪ cuni 4830   ↦ cmpt 5137  ⟶wf 6344  ‘cfv 6348  ℩crio 7105  CHOICEwac 9533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-riota 7106  df-ac 9534

This theorem is referenced by:  dfac2  9549  axac2  9880