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Theorem dfacacn 8914
 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 3192 . . . 4 𝑥 ∈ V
2 acacni 8913 . . . 4 ((CHOICE𝑥 ∈ V) → AC 𝑥 = V)
31, 2mpan2 706 . . 3 (CHOICEAC 𝑥 = V)
43alrimiv 1852 . 2 (CHOICE → ∀𝑥AC 𝑥 = V)
5 vex 3192 . . . . . . 7 𝑦 ∈ V
6 difexg 4773 . . . . . . 7 (𝑦 ∈ V → (𝑦 ∖ {∅}) ∈ V)
75, 6ax-mp 5 . . . . . 6 (𝑦 ∖ {∅}) ∈ V
8 acneq 8817 . . . . . . 7 (𝑥 = (𝑦 ∖ {∅}) → AC 𝑥 = AC (𝑦 ∖ {∅}))
98eqeq1d 2623 . . . . . 6 (𝑥 = (𝑦 ∖ {∅}) → (AC 𝑥 = V ↔ AC (𝑦 ∖ {∅}) = V))
107, 9spcv 3288 . . . . 5 (∀𝑥AC 𝑥 = V → AC (𝑦 ∖ {∅}) = V)
11 vuniex 6914 . . . . . . 7 𝑦 ∈ V
12 id 22 . . . . . . 7 (AC (𝑦 ∖ {∅}) = V → AC (𝑦 ∖ {∅}) = V)
1311, 12syl5eleqr 2705 . . . . . 6 (AC (𝑦 ∖ {∅}) = V → 𝑦AC (𝑦 ∖ {∅}))
14 eldifi 3715 . . . . . . . . 9 (𝑧 ∈ (𝑦 ∖ {∅}) → 𝑧𝑦)
15 elssuni 4438 . . . . . . . . 9 (𝑧𝑦𝑧 𝑦)
1614, 15syl 17 . . . . . . . 8 (𝑧 ∈ (𝑦 ∖ {∅}) → 𝑧 𝑦)
17 eldifsni 4294 . . . . . . . 8 (𝑧 ∈ (𝑦 ∖ {∅}) → 𝑧 ≠ ∅)
1816, 17jca 554 . . . . . . 7 (𝑧 ∈ (𝑦 ∖ {∅}) → (𝑧 𝑦𝑧 ≠ ∅))
1918rgen 2917 . . . . . 6 𝑧 ∈ (𝑦 ∖ {∅})(𝑧 𝑦𝑧 ≠ ∅)
20 acni2 8820 . . . . . 6 (( 𝑦AC (𝑦 ∖ {∅}) ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑧 𝑦𝑧 ≠ ∅)) → ∃𝑔(𝑔:(𝑦 ∖ {∅})⟶ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔𝑧) ∈ 𝑧))
2113, 19, 20sylancl 693 . . . . 5 (AC (𝑦 ∖ {∅}) = V → ∃𝑔(𝑔:(𝑦 ∖ {∅})⟶ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔𝑧) ∈ 𝑧))
225mptex 6446 . . . . . . 7 (𝑥𝑦 ↦ (𝑔𝑥)) ∈ V
23 simpr 477 . . . . . . . . 9 ((𝑔:(𝑦 ∖ {∅})⟶ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔𝑧) ∈ 𝑧) → ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔𝑧) ∈ 𝑧)
24 eldifsn 4292 . . . . . . . . . . . 12 (𝑧 ∈ (𝑦 ∖ {∅}) ↔ (𝑧𝑦𝑧 ≠ ∅))
2524imbi1i 339 . . . . . . . . . . 11 ((𝑧 ∈ (𝑦 ∖ {∅}) → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧) ↔ ((𝑧𝑦𝑧 ≠ ∅) → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧))
26 fveq2 6153 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝑔𝑥) = (𝑔𝑧))
27 eqid 2621 . . . . . . . . . . . . . . 15 (𝑥𝑦 ↦ (𝑔𝑥)) = (𝑥𝑦 ↦ (𝑔𝑥))
28 fvex 6163 . . . . . . . . . . . . . . 15 (𝑔𝑧) ∈ V
2926, 27, 28fvmpt 6244 . . . . . . . . . . . . . 14 (𝑧𝑦 → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) = (𝑔𝑧))
3014, 29syl 17 . . . . . . . . . . . . 13 (𝑧 ∈ (𝑦 ∖ {∅}) → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) = (𝑔𝑧))
3130eleq1d 2683 . . . . . . . . . . . 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 462 . . . . . . . . . . 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 2973 . . . . . . . . 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 6178 . . . . . . . . . . 11 (𝑔𝑥) ∈ (ran 𝑔 ∪ {∅})
3837rgenw 2919 . . . . . . . . . 10 𝑥𝑦 (𝑔𝑥) ∈ (ran 𝑔 ∪ {∅})
3927fmpt 6342 . . . . . . . . . 10 (∀𝑥𝑦 (𝑔𝑥) ∈ (ran 𝑔 ∪ {∅}) ↔ (𝑥𝑦 ↦ (𝑔𝑥)):𝑦⟶(ran 𝑔 ∪ {∅}))
4038, 39mpbi 220 . . . . . . . . 9 (𝑥𝑦 ↦ (𝑔𝑥)):𝑦⟶(ran 𝑔 ∪ {∅})
41 ffn 6007 . . . . . . . . 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 5942 . . . . . . . . 9 (𝑓 = (𝑥𝑦 ↦ (𝑔𝑥)) → (𝑓 Fn 𝑦 ↔ (𝑥𝑦 ↦ (𝑔𝑥)) Fn 𝑦))
45 fveq1 6152 . . . . . . . . . . . 12 (𝑓 = (𝑥𝑦 ↦ (𝑔𝑥)) → (𝑓𝑧) = ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧))
4645eleq1d 2683 . . . . . . . . . . 11 (𝑓 = (𝑥𝑦 ↦ (𝑔𝑥)) → ((𝑓𝑧) ∈ 𝑧 ↔ ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧))
4746imbi2d 330 . . . . . . . . . 10 (𝑓 = (𝑥𝑦 ↦ (𝑔𝑥)) → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧)))
4847ralbidv 2981 . . . . . . . . 9 (𝑓 = (𝑥𝑦 ↦ (𝑔𝑥)) → (∀𝑧𝑦 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑧𝑦 (𝑧 ≠ ∅ → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧)))
4944, 48anbi12d 746 . . . . . . . 8 (𝑓 = (𝑥𝑦 ↦ (𝑔𝑥)) → ((𝑓 Fn 𝑦 ∧ ∀𝑧𝑦 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) ↔ ((𝑥𝑦 ↦ (𝑔𝑥)) Fn 𝑦 ∧ ∀𝑧𝑦 (𝑧 ≠ ∅ → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧))))
5049spcegv 3283 . . . . . . 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 1855 . . . . 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 1852 . . 3 (∀𝑥AC 𝑥 = V → ∀𝑦𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧𝑦 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
55 dfac4 8896 . . 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 384  ∀wal 1478   = wceq 1480  ∃wex 1701   ∈ wcel 1987   ≠ wne 2790  ∀wral 2907  Vcvv 3189   ∖ cdif 3556   ∪ cun 3557   ⊆ wss 3559  ∅c0 3896  {csn 4153  ∪ cuni 4407   ↦ cmpt 4678  ran crn 5080   Fn wfn 5847  ⟶wf 5848  ‘cfv 5852  AC wacn 8715  CHOICEwac 8889 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-er 7694  df-map 7811  df-en 7907  df-dom 7908  df-card 8716  df-acn 8719  df-ac 8890 This theorem is referenced by:  dfac13  8915
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