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Theorem dfacacn 9615
 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 acacni 9614 . . . 4 ((CHOICE𝑥 ∈ V) → AC 𝑥 = V)
21elvd 3417 . . 3 (CHOICEAC 𝑥 = V)
32alrimiv 1929 . 2 (CHOICE → ∀𝑥AC 𝑥 = V)
4 vex 3414 . . . . . . 7 𝑦 ∈ V
54difexi 5203 . . . . . 6 (𝑦 ∖ {∅}) ∈ V
6 acneq 9517 . . . . . . 7 (𝑥 = (𝑦 ∖ {∅}) → AC 𝑥 = AC (𝑦 ∖ {∅}))
76eqeq1d 2761 . . . . . 6 (𝑥 = (𝑦 ∖ {∅}) → (AC 𝑥 = V ↔ AC (𝑦 ∖ {∅}) = V))
85, 7spcv 3527 . . . . 5 (∀𝑥AC 𝑥 = V → AC (𝑦 ∖ {∅}) = V)
9 vuniex 7470 . . . . . . 7 𝑦 ∈ V
10 id 22 . . . . . . 7 (AC (𝑦 ∖ {∅}) = V → AC (𝑦 ∖ {∅}) = V)
119, 10eleqtrrid 2860 . . . . . 6 (AC (𝑦 ∖ {∅}) = V → 𝑦AC (𝑦 ∖ {∅}))
12 eldifi 4035 . . . . . . . . 9 (𝑧 ∈ (𝑦 ∖ {∅}) → 𝑧𝑦)
13 elssuni 4834 . . . . . . . . 9 (𝑧𝑦𝑧 𝑦)
1412, 13syl 17 . . . . . . . 8 (𝑧 ∈ (𝑦 ∖ {∅}) → 𝑧 𝑦)
15 eldifsni 4684 . . . . . . . 8 (𝑧 ∈ (𝑦 ∖ {∅}) → 𝑧 ≠ ∅)
1614, 15jca 515 . . . . . . 7 (𝑧 ∈ (𝑦 ∖ {∅}) → (𝑧 𝑦𝑧 ≠ ∅))
1716rgen 3081 . . . . . 6 𝑧 ∈ (𝑦 ∖ {∅})(𝑧 𝑦𝑧 ≠ ∅)
18 acni2 9520 . . . . . 6 (( 𝑦AC (𝑦 ∖ {∅}) ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑧 𝑦𝑧 ≠ ∅)) → ∃𝑔(𝑔:(𝑦 ∖ {∅})⟶ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔𝑧) ∈ 𝑧))
1911, 17, 18sylancl 589 . . . . 5 (AC (𝑦 ∖ {∅}) = V → ∃𝑔(𝑔:(𝑦 ∖ {∅})⟶ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔𝑧) ∈ 𝑧))
204mptex 6984 . . . . . . 7 (𝑥𝑦 ↦ (𝑔𝑥)) ∈ V
21 simpr 488 . . . . . . . . 9 ((𝑔:(𝑦 ∖ {∅})⟶ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔𝑧) ∈ 𝑧) → ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔𝑧) ∈ 𝑧)
22 eldifsn 4681 . . . . . . . . . . . 12 (𝑧 ∈ (𝑦 ∖ {∅}) ↔ (𝑧𝑦𝑧 ≠ ∅))
2322imbi1i 353 . . . . . . . . . . 11 ((𝑧 ∈ (𝑦 ∖ {∅}) → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧) ↔ ((𝑧𝑦𝑧 ≠ ∅) → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧))
24 fveq2 6664 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → (𝑔𝑥) = (𝑔𝑧))
25 eqid 2759 . . . . . . . . . . . . . . 15 (𝑥𝑦 ↦ (𝑔𝑥)) = (𝑥𝑦 ↦ (𝑔𝑥))
26 fvex 6677 . . . . . . . . . . . . . . 15 (𝑔𝑧) ∈ V
2724, 25, 26fvmpt 6765 . . . . . . . . . . . . . 14 (𝑧𝑦 → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) = (𝑔𝑧))
2812, 27syl 17 . . . . . . . . . . . . 13 (𝑧 ∈ (𝑦 ∖ {∅}) → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) = (𝑔𝑧))
2928eleq1d 2837 . . . . . . . . . . . 12 (𝑧 ∈ (𝑦 ∖ {∅}) → (((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧 ↔ (𝑔𝑧) ∈ 𝑧))
3029pm5.74i 274 . . . . . . . . . . 11 ((𝑧 ∈ (𝑦 ∖ {∅}) → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧) ↔ (𝑧 ∈ (𝑦 ∖ {∅}) → (𝑔𝑧) ∈ 𝑧))
31 impexp 454 . . . . . . . . . . 11 (((𝑧𝑦𝑧 ≠ ∅) → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧) ↔ (𝑧𝑦 → (𝑧 ≠ ∅ → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧)))
3223, 30, 313bitr3i 304 . . . . . . . . . 10 ((𝑧 ∈ (𝑦 ∖ {∅}) → (𝑔𝑧) ∈ 𝑧) ↔ (𝑧𝑦 → (𝑧 ≠ ∅ → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧)))
3332ralbii2 3096 . . . . . . . . 9 (∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔𝑧) ∈ 𝑧 ↔ ∀𝑧𝑦 (𝑧 ≠ ∅ → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧))
3421, 33sylib 221 . . . . . . . 8 ((𝑔:(𝑦 ∖ {∅})⟶ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔𝑧) ∈ 𝑧) → ∀𝑧𝑦 (𝑧 ≠ ∅ → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧))
35 fvrn0 6692 . . . . . . . . . . 11 (𝑔𝑥) ∈ (ran 𝑔 ∪ {∅})
3635rgenw 3083 . . . . . . . . . 10 𝑥𝑦 (𝑔𝑥) ∈ (ran 𝑔 ∪ {∅})
3725fmpt 6872 . . . . . . . . . 10 (∀𝑥𝑦 (𝑔𝑥) ∈ (ran 𝑔 ∪ {∅}) ↔ (𝑥𝑦 ↦ (𝑔𝑥)):𝑦⟶(ran 𝑔 ∪ {∅}))
3836, 37mpbi 233 . . . . . . . . 9 (𝑥𝑦 ↦ (𝑔𝑥)):𝑦⟶(ran 𝑔 ∪ {∅})
39 ffn 6504 . . . . . . . . 9 ((𝑥𝑦 ↦ (𝑔𝑥)):𝑦⟶(ran 𝑔 ∪ {∅}) → (𝑥𝑦 ↦ (𝑔𝑥)) Fn 𝑦)
4038, 39ax-mp 5 . . . . . . . 8 (𝑥𝑦 ↦ (𝑔𝑥)) Fn 𝑦
4134, 40jctil 523 . . . . . . 7 ((𝑔:(𝑦 ∖ {∅})⟶ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔𝑧) ∈ 𝑧) → ((𝑥𝑦 ↦ (𝑔𝑥)) Fn 𝑦 ∧ ∀𝑧𝑦 (𝑧 ≠ ∅ → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧)))
42 fneq1 6431 . . . . . . . . 9 (𝑓 = (𝑥𝑦 ↦ (𝑔𝑥)) → (𝑓 Fn 𝑦 ↔ (𝑥𝑦 ↦ (𝑔𝑥)) Fn 𝑦))
43 fveq1 6663 . . . . . . . . . . . 12 (𝑓 = (𝑥𝑦 ↦ (𝑔𝑥)) → (𝑓𝑧) = ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧))
4443eleq1d 2837 . . . . . . . . . . 11 (𝑓 = (𝑥𝑦 ↦ (𝑔𝑥)) → ((𝑓𝑧) ∈ 𝑧 ↔ ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧))
4544imbi2d 344 . . . . . . . . . 10 (𝑓 = (𝑥𝑦 ↦ (𝑔𝑥)) → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧)))
4645ralbidv 3127 . . . . . . . . 9 (𝑓 = (𝑥𝑦 ↦ (𝑔𝑥)) → (∀𝑧𝑦 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑧𝑦 (𝑧 ≠ ∅ → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧)))
4742, 46anbi12d 633 . . . . . . . 8 (𝑓 = (𝑥𝑦 ↦ (𝑔𝑥)) → ((𝑓 Fn 𝑦 ∧ ∀𝑧𝑦 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) ↔ ((𝑥𝑦 ↦ (𝑔𝑥)) Fn 𝑦 ∧ ∀𝑧𝑦 (𝑧 ≠ ∅ → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧))))
4847spcegv 3518 . . . . . . 7 ((𝑥𝑦 ↦ (𝑔𝑥)) ∈ V → (((𝑥𝑦 ↦ (𝑔𝑥)) Fn 𝑦 ∧ ∀𝑧𝑦 (𝑧 ≠ ∅ → ((𝑥𝑦 ↦ (𝑔𝑥))‘𝑧) ∈ 𝑧)) → ∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧𝑦 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))))
4920, 41, 48mpsyl 68 . . . . . 6 ((𝑔:(𝑦 ∖ {∅})⟶ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔𝑧) ∈ 𝑧) → ∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧𝑦 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
5049exlimiv 1932 . . . . 5 (∃𝑔(𝑔:(𝑦 ∖ {∅})⟶ 𝑦 ∧ ∀𝑧 ∈ (𝑦 ∖ {∅})(𝑔𝑧) ∈ 𝑧) → ∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧𝑦 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
518, 19, 503syl 18 . . . 4 (∀𝑥AC 𝑥 = V → ∃𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧𝑦 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
5251alrimiv 1929 . . 3 (∀𝑥AC 𝑥 = V → ∀𝑦𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧𝑦 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
53 dfac4 9596 . . 3 (CHOICE ↔ ∀𝑦𝑓(𝑓 Fn 𝑦 ∧ ∀𝑧𝑦 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
5452, 53sylibr 237 . 2 (∀𝑥AC 𝑥 = V → CHOICE)
553, 54impbii 212 1 (CHOICE ↔ ∀𝑥AC 𝑥 = V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2112   ≠ wne 2952  ∀wral 3071  Vcvv 3410   ∖ cdif 3858   ∪ cun 3859   ⊆ wss 3861  ∅c0 4228  {csn 4526  ∪ cuni 4802   ↦ cmpt 5117  ran crn 5530   Fn wfn 6336  ⟶wf 6337  ‘cfv 6341  AC wacn 9414  CHOICEwac 9589 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5161  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303  ax-un 7466 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-pss 3880  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4803  df-int 4843  df-iun 4889  df-br 5038  df-opab 5100  df-mpt 5118  df-tr 5144  df-id 5435  df-eprel 5440  df-po 5448  df-so 5449  df-fr 5488  df-se 5489  df-we 5490  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-pred 6132  df-ord 6178  df-on 6179  df-suc 6181  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-isom 6350  df-riota 7115  df-ov 7160  df-oprab 7161  df-mpo 7162  df-1st 7700  df-2nd 7701  df-wrecs 7964  df-recs 8025  df-er 8306  df-map 8425  df-en 8542  df-dom 8543  df-card 9415  df-acn 9418  df-ac 9590 This theorem is referenced by:  dfac13  9616
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