Theorem dfacacn 10100

Description: A choice equivalent: every set has choice sets of every length. (Contributed by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
dfacacn	⊢ (CHOICE ↔ ∀𝑥AC 𝑥 = V)

Proof of Theorem dfacacn

Dummy variables 𝑓 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		acacni 10099	. . . 4 ⊢ ((CHOICE ∧ 𝑥 ∈ V) → AC 𝑥 = V)
2	1	elvd 3462	. . 3 ⊢ (CHOICE → AC 𝑥 = V)
3	2	alrimiv 1949	. 2 ⊢ (CHOICE → ∀𝑥AC 𝑥 = V)
4		vex 3460	. . . . . . 7 ⊢ 𝑦 ∈ V
5	4	difexi 5288	. . . . . 6 ⊢ (𝑦 ∖ {∅}) ∈ V
6		acneq 10001	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∖ {∅}) → AC 𝑥 = AC (𝑦 ∖ {∅}))
7	6	eqeq1d 2766	. . . . . 6 ⊢ (𝑥 = (𝑦 ∖ {∅}) → (AC 𝑥 = V ↔ AC (𝑦 ∖ {∅}) = V))
8	5, 7	spcv 3566	. . . . 5 ⊢ (∀𝑥AC 𝑥 = V → AC (𝑦 ∖ {∅}) = V)
9		vuniex 7724	. . . . . . 7 ⊢ ∪ 𝑦 ∈ V
10		id 22	. . . . . . 7 ⊢ (AC (𝑦 ∖ {∅}) = V → AC (𝑦 ∖ {∅}) = V)
11	9, 10	eleqtrrid 2871	. . . . . 6 ⊢ (AC (𝑦 ∖ {∅}) = V → ∪ 𝑦 ∈ AC (𝑦 ∖ {∅}))
12		eldifi 4086	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → 𝑧 ∈ 𝑦)
13		elssuni 4899	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑦 → 𝑧 ⊆ ∪ 𝑦)
14	12, 13	syl 17	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → 𝑧 ⊆ ∪ 𝑦)
15		eldifsni 4752	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → 𝑧 ≠ ∅)
16	14, 15	jca 519	. . . . . . 7 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → (𝑧 ⊆ ∪ 𝑦 ∧ 𝑧 ≠ ∅))
17	16	rgen 3080	. . . . . 6 ⊢ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑧 ⊆ ∪ 𝑦 ∧ 𝑧 ≠ ∅)
18		acni2 10004	. . . . . 6 ⊢ ((∪ 𝑦 ∈ AC (𝑦 ∖ {∅}) ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑧 ⊆ ∪ 𝑦 ∧ 𝑧 ≠ ∅)) → ∃𝑔(𝑔:(𝑦 ∖ {∅})⟶∪ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧))
19	11, 17, 18	sylancl 595	. . . . 5 ⊢ (AC (𝑦 ∖ {∅}) = V → ∃𝑔(𝑔:(𝑦 ∖ {∅})⟶∪ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧))
20	4	mptex 7209	. . . . . . 7 ⊢ (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) ∈ V
21		eldifsn 4748	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) ↔ (𝑧 ∈ 𝑦 ∧ 𝑧 ≠ ∅))
22	21	imbi1i 351	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (𝑦 ∖ {∅}) → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧) ↔ ((𝑧 ∈ 𝑦 ∧ 𝑧 ≠ ∅) → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧))
23		fveq2 6869	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (𝑔‘𝑥) = (𝑔‘𝑧))
24		eqid 2764	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))
25		fvex 6882	. . . . . . . . . . . . . . 15 ⊢ (𝑔‘𝑧) ∈ V
26	23, 24, 25	fvmpt 6977	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑦 → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) = (𝑔‘𝑧))
27	12, 26	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) = (𝑔‘𝑧))
28	27	eleq1d 2849	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑦 ∖ {∅}) → (((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧 ↔ (𝑔‘𝑧) ∈ 𝑧))
29	28	pm5.74i 273	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (𝑦 ∖ {∅}) → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧) ↔ (𝑧 ∈ (𝑦 ∖ {∅}) → (𝑔‘𝑧) ∈ 𝑧))
30		impexp 454	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ 𝑦 ∧ 𝑧 ≠ ∅) → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧) ↔ (𝑧 ∈ 𝑦 → (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧)))
31	22, 29, 30	3bitr3i 303	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (𝑦 ∖ {∅}) → (𝑔‘𝑧) ∈ 𝑧) ↔ (𝑧 ∈ 𝑦 → (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧)))
32	31	ralbii2 3106	. . . . . . . . 9 ⊢ (∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧 ↔ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧))
33	32	bilani 508	. . . . . . . 8 ⊢ ((𝑔:(𝑦 ∖ {∅})⟶∪ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧) → ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧))
34		fvrn0 6897	. . . . . . . . . . 11 ⊢ (𝑔‘𝑥) ∈ (ran 𝑔 ∪ {∅})
35	34	rgenw 3082	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ 𝑦 (𝑔‘𝑥) ∈ (ran 𝑔 ∪ {∅})
36	24	fmpt 7093	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑦 (𝑔‘𝑥) ∈ (ran 𝑔 ∪ {∅}) ↔ (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)):𝑦⟶(ran 𝑔 ∪ {∅}))
37	35, 36	mpbi 232	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)):𝑦⟶(ran 𝑔 ∪ {∅})
38		ffn 6693	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)):𝑦⟶(ran 𝑔 ∪ {∅}) → (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) Fn 𝑦)
39	37, 38	ax-mp 5	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) Fn 𝑦
40	33, 39	jctil 527	. . . . . . 7 ⊢ ((𝑔:(𝑦 ∖ {∅})⟶∪ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧) → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧)))
41		fneq1 6614	. . . . . . . . 9 ⊢ (𝑓 = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) → (𝑓 Fn 𝑦 ↔ (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) Fn 𝑦))
42		fveq1 6868	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) → (𝑓‘𝑧) = ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧))
43	42	eleq1d 2849	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) → ((𝑓‘𝑧) ∈ 𝑧 ↔ ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧))
44	43	imbi2d 342	. . . . . . . . . 10 ⊢ (𝑓 = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) → ((𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧)))
45	44	ralbidv 3187	. . . . . . . . 9 ⊢ (𝑓 = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) → (∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧) ↔ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧)))
46	41, 45	anbi12d 641	. . . . . . . 8 ⊢ (𝑓 = (𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) → ((𝑓 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)) ↔ ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧))))
47	46	spcegv 3558	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) ∈ V → (((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥)) Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → ((𝑥 ∈ 𝑦 ↦ (𝑔‘𝑥))‘𝑧) ∈ 𝑧)) → ∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))))
48	20, 40, 47	mpsyl 68	. . . . . 6 ⊢ ((𝑔:(𝑦 ∖ {∅})⟶∪ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧) → ∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
49	48	exlimiv 1952	. . . . 5 ⊢ (∃𝑔(𝑔:(𝑦 ∖ {∅})⟶∪ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔‘𝑧) ∈ 𝑧) → ∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
50	8, 19, 49	3syl 18	. . . 4 ⊢ (∀𝑥AC 𝑥 = V → ∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
51	50	alrimiv 1949	. . 3 ⊢ (∀𝑥AC 𝑥 = V → ∀𝑦∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
52		dfac4 10080	. . 3 ⊢ (CHOICE ↔ ∀𝑦∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧)))
53	51, 52	sylibr 236	. 2 ⊢ (∀𝑥AC 𝑥 = V → CHOICE)
54	3, 53	impbii 211	1 ⊢ (CHOICE ↔ ∀𝑥AC 𝑥 = V)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1560   = wceq 1562  ∃wex 1801   ∈ wcel 2144   ≠ wne 2959  ∀wral 3078  Vcvv 3456   ∖ cdif 3903   ∪ cun 3904   ⊆ wss 3906  ∅c0 4287  {csn 4584  ∪ cuni 4867   ↦ cmpt 5183  ran crn 5650   Fn wfn 6518  ⟶wf 6519  ‘cfv 6523  AC wacn 9898  CHOICEwac 10073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-se 5603  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-isom 6532  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-er 8680  df-map 8812  df-en 8930  df-dom 8931  df-card 9899  df-acn 9902  df-ac 10074

This theorem is referenced by:  dfac13  10101