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Theorem dfacacn 9001
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.
StepHypRef Expression
1 vex 3234 . . . 4 𝑥 ∈ V
2 acacni 9000 . . . 4 ((CHOICE𝑥 ∈ V) → AC 𝑥 = V)
31, 2mpan2 707 . . 3 (CHOICEAC 𝑥 = V)
43alrimiv 1895 . 2 (CHOICE → ∀𝑥AC 𝑥 = V)
5 vex 3234 . . . . . . 7 𝑦 ∈ V
6 difexg 4841 . . . . . . 7 (𝑦 ∈ V → (𝑦 ∖ {∅}) ∈ V)
75, 6ax-mp 5 . . . . . 6 (𝑦 ∖ {∅}) ∈ V
8 acneq 8904 . . . . . . 7 (𝑥 = (𝑦 ∖ {∅}) → AC 𝑥 = AC (𝑦 ∖ {∅}))
98eqeq1d 2653 . . . . . 6 (𝑥 = (𝑦 ∖ {∅}) → (AC 𝑥 = V ↔ AC (𝑦 ∖ {∅}) = V))
107, 9spcv 3330 . . . . 5 (∀𝑥AC 𝑥 = V → AC (𝑦 ∖ {∅}) = V)
11 vuniex 6996 . . . . . . 7 𝑦 ∈ V
12 id 22 . . . . . . 7 (AC (𝑦 ∖ {∅}) = V → AC (𝑦 ∖ {∅}) = V)
1311, 12syl5eleqr 2737 . . . . . 6 (AC (𝑦 ∖ {∅}) = V → 𝑦AC (𝑦 ∖ {∅}))
14 eldifi 3765 . . . . . . . . 9 (𝑧 ∈ (𝑦 ∖ {∅}) → 𝑧𝑦)
15 elssuni 4499 . . . . . . . . 9 (𝑧𝑦𝑧 𝑦)
1614, 15syl 17 . . . . . . . 8 (𝑧 ∈ (𝑦 ∖ {∅}) → 𝑧 𝑦)
17 eldifsni 4353 . . . . . . . 8 (𝑧 ∈ (𝑦 ∖ {∅}) → 𝑧 ≠ ∅)
1816, 17jca 553 . . . . . . 7 (𝑧 ∈ (𝑦 ∖ {∅}) → (𝑧 𝑦𝑧 ≠ ∅))
1918rgen 2951 . . . . . 6 𝑧 ∈ (𝑦 ∖ {∅})(𝑧 𝑦𝑧 ≠ ∅)
20 acni2 8907 . . . . . 6 (( 𝑦AC (𝑦 ∖ {∅}) ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑧 𝑦𝑧 ≠ ∅)) → ∃𝑔(𝑔:(𝑦 ∖ {∅})⟶ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔𝑧) ∈ 𝑧))
2113, 19, 20sylancl 695 . . . . 5 (AC (𝑦 ∖ {∅}) = V → ∃𝑔(𝑔:(𝑦 ∖ {∅})⟶ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔𝑧) ∈ 𝑧))
225mptex 6527 . . . . . . 7 (𝑥𝑦 ↦ (𝑔𝑥)) ∈ V
23 simpr 476 . . . . . . . . 9 ((𝑔:(𝑦 ∖ {∅})⟶ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔𝑧) ∈ 𝑧) → ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔𝑧) ∈ 𝑧)
24 eldifsn 4350 . . . . . . . . . . . 12 (𝑧 ∈ (𝑦 ∖ {∅}) ↔ (𝑧𝑦𝑧 ≠ ∅))
2524imbi1i 338 . . . . . . . . . . 11 ((𝑧 ∈ (𝑦 ∖ {∅}) → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧) ↔ ((𝑧𝑦𝑧 ≠ ∅) → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧))
26 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝑔𝑥) = (𝑔𝑧))
27 eqid 2651 . . . . . . . . . . . . . . 15 (𝑥𝑦 ↦ (𝑔𝑥)) = (𝑥𝑦 ↦ (𝑔𝑥))
28 fvex 6239 . . . . . . . . . . . . . . 15 (𝑔𝑧) ∈ V
2926, 27, 28fvmpt 6321 . . . . . . . . . . . . . 14 (𝑧𝑦 → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) = (𝑔𝑧))
3014, 29syl 17 . . . . . . . . . . . . 13 (𝑧 ∈ (𝑦 ∖ {∅}) → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) = (𝑔𝑧))
3130eleq1d 2715 . . . . . . . . . . . 12 (𝑧 ∈ (𝑦 ∖ {∅}) → (((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧 ↔ (𝑔𝑧) ∈ 𝑧))
3231pm5.74i 260 . . . . . . . . . . 11 ((𝑧 ∈ (𝑦 ∖ {∅}) → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧) ↔ (𝑧 ∈ (𝑦 ∖ {∅}) → (𝑔𝑧) ∈ 𝑧))
33 impexp 461 . . . . . . . . . . 11 (((𝑧𝑦𝑧 ≠ ∅) → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧) ↔ (𝑧𝑦 → (𝑧 ≠ ∅ → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧)))
3425, 32, 333bitr3i 290 . . . . . . . . . 10 ((𝑧 ∈ (𝑦 ∖ {∅}) → (𝑔𝑧) ∈ 𝑧) ↔ (𝑧𝑦 → (𝑧 ≠ ∅ → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧)))
3534ralbii2 3007 . . . . . . . . 9 (∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔𝑧) ∈ 𝑧 ↔ ∀𝑧𝑦 (𝑧 ≠ ∅ → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧))
3623, 35sylib 208 . . . . . . . 8 ((𝑔:(𝑦 ∖ {∅})⟶ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔𝑧) ∈ 𝑧) → ∀𝑧𝑦 (𝑧 ≠ ∅ → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧))
37 fvrn0 6254 . . . . . . . . . . 11 (𝑔𝑥) ∈ (ran 𝑔 ∪ {∅})
3837rgenw 2953 . . . . . . . . . 10 𝑥𝑦 (𝑔𝑥) ∈ (ran 𝑔 ∪ {∅})
3927fmpt 6421 . . . . . . . . . 10 (∀𝑥𝑦 (𝑔𝑥) ∈ (ran 𝑔 ∪ {∅}) ↔ (𝑥𝑦 ↦ (𝑔𝑥)):𝑦⟶(ran 𝑔 ∪ {∅}))
4038, 39mpbi 220 . . . . . . . . 9 (𝑥𝑦 ↦ (𝑔𝑥)):𝑦⟶(ran 𝑔 ∪ {∅})
41 ffn 6083 . . . . . . . . 9 ((𝑥𝑦 ↦ (𝑔𝑥)):𝑦⟶(ran 𝑔 ∪ {∅}) → (𝑥𝑦 ↦ (𝑔𝑥)) Fn 𝑦)
4240, 41ax-mp 5 . . . . . . . 8 (𝑥𝑦 ↦ (𝑔𝑥)) Fn 𝑦
4336, 42jctil 559 . . . . . . 7 ((𝑔:(𝑦 ∖ {∅})⟶ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔𝑧) ∈ 𝑧) → ((𝑥𝑦 ↦ (𝑔𝑥)) Fn 𝑦 ∧ ∀𝑧𝑦 (𝑧 ≠ ∅ → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧)))
44 fneq1 6017 . . . . . . . . 9 (𝑓 = (𝑥𝑦 ↦ (𝑔𝑥)) → (𝑓 Fn 𝑦 ↔ (𝑥𝑦 ↦ (𝑔𝑥)) Fn 𝑦))
45 fveq1 6228 . . . . . . . . . . . 12 (𝑓 = (𝑥𝑦 ↦ (𝑔𝑥)) → (𝑓𝑧) = ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧))
4645eleq1d 2715 . . . . . . . . . . 11 (𝑓 = (𝑥𝑦 ↦ (𝑔𝑥)) → ((𝑓𝑧) ∈ 𝑧 ↔ ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧))
4746imbi2d 329 . . . . . . . . . 10 (𝑓 = (𝑥𝑦 ↦ (𝑔𝑥)) → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧)))
4847ralbidv 3015 . . . . . . . . 9 (𝑓 = (𝑥𝑦 ↦ (𝑔𝑥)) → (∀𝑧𝑦 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑧𝑦 (𝑧 ≠ ∅ → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧)))
4944, 48anbi12d 747 . . . . . . . 8 (𝑓 = (𝑥𝑦 ↦ (𝑔𝑥)) → ((𝑓 Fn 𝑦 ∧ ∀𝑧𝑦 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) ↔ ((𝑥𝑦 ↦ (𝑔𝑥)) Fn 𝑦 ∧ ∀𝑧𝑦 (𝑧 ≠ ∅ → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧))))
5049spcegv 3325 . . . . . . 7 ((𝑥𝑦 ↦ (𝑔𝑥)) ∈ V → (((𝑥𝑦 ↦ (𝑔𝑥)) Fn 𝑦 ∧ ∀𝑧𝑦 (𝑧 ≠ ∅ → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧)) → ∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧𝑦 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))))
5122, 43, 50mpsyl 68 . . . . . 6 ((𝑔:(𝑦 ∖ {∅})⟶ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔𝑧) ∈ 𝑧) → ∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧𝑦 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
5251exlimiv 1898 . . . . 5 (∃𝑔(𝑔:(𝑦 ∖ {∅})⟶ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔𝑧) ∈ 𝑧) → ∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧𝑦 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
5310, 21, 523syl 18 . . . 4 (∀𝑥AC 𝑥 = V → ∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧𝑦 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
5453alrimiv 1895 . . 3 (∀𝑥AC 𝑥 = V → ∀𝑦𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧𝑦 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
55 dfac4 8983 . . 3 (CHOICE ↔ ∀𝑦𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧𝑦 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
5654, 55sylibr 224 . 2 (∀𝑥AC 𝑥 = V → CHOICE)
574, 56impbii 199 1 (CHOICE ↔ ∀𝑥AC 𝑥 = V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wex 1744  wcel 2030  wne 2823  wral 2941  Vcvv 3231  cdif 3604  cun 3605  wss 3607  c0 3948  {csn 4210   cuni 4468  cmpt 4762  ran crn 5144   Fn wfn 5921  wf 5922  cfv 5926  AC wacn 8802  CHOICEwac 8976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-card 8803  df-acn 8806  df-ac 8977
This theorem is referenced by:  dfac13  9002
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