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Theorem dfac4 9546
Description: Equivalence of two versions of the Axiom of Choice. The right-hand side is Axiom AC of [BellMachover] p. 488. The proof does not depend on AC. (Contributed by NM, 24-Mar-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)
Assertion
Ref Expression
dfac4 (CHOICE ↔ ∀𝑥𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
Distinct variable group:   𝑥,𝑓,𝑧

Proof of Theorem dfac4
Dummy variables 𝑦 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfac3 9545 . 2 (CHOICE ↔ ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
2 fveq1 6660 . . . . . . . . 9 (𝑓 = 𝑦 → (𝑓𝑧) = (𝑦𝑧))
32eleq1d 2900 . . . . . . . 8 (𝑓 = 𝑦 → ((𝑓𝑧) ∈ 𝑧 ↔ (𝑦𝑧) ∈ 𝑧))
43imbi2d 344 . . . . . . 7 (𝑓 = 𝑦 → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧)))
54ralbidv 3192 . . . . . 6 (𝑓 = 𝑦 → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧)))
65cbvexvw 2045 . . . . 5 (∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧))
7 fvex 6674 . . . . . . . . 9 (𝑦𝑤) ∈ V
8 eqid 2824 . . . . . . . . 9 (𝑤𝑥 ↦ (𝑦𝑤)) = (𝑤𝑥 ↦ (𝑦𝑤))
97, 8fnmpti 6480 . . . . . . . 8 (𝑤𝑥 ↦ (𝑦𝑤)) Fn 𝑥
10 fveq2 6661 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (𝑦𝑤) = (𝑦𝑧))
11 fvex 6674 . . . . . . . . . . . . 13 (𝑦𝑧) ∈ V
1210, 8, 11fvmpt 6759 . . . . . . . . . . . 12 (𝑧𝑥 → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) = (𝑦𝑧))
1312eleq1d 2900 . . . . . . . . . . 11 (𝑧𝑥 → (((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧 ↔ (𝑦𝑧) ∈ 𝑧))
1413imbi2d 344 . . . . . . . . . 10 (𝑧𝑥 → ((𝑧 ≠ ∅ → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧)))
1514ralbiia 3159 . . . . . . . . 9 (∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧))
1615anbi2i 625 . . . . . . . 8 (((𝑤𝑥 ↦ (𝑦𝑤)) Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧)) ↔ ((𝑤𝑥 ↦ (𝑦𝑤)) Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧)))
179, 16mpbiran 708 . . . . . . 7 (((𝑤𝑥 ↦ (𝑦𝑤)) Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧)) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧))
18 fvrn0 6689 . . . . . . . . . . 11 (𝑦𝑤) ∈ (ran 𝑦 ∪ {∅})
1918rgenw 3145 . . . . . . . . . 10 𝑤𝑥 (𝑦𝑤) ∈ (ran 𝑦 ∪ {∅})
208fmpt 6865 . . . . . . . . . 10 (∀𝑤𝑥 (𝑦𝑤) ∈ (ran 𝑦 ∪ {∅}) ↔ (𝑤𝑥 ↦ (𝑦𝑤)):𝑥⟶(ran 𝑦 ∪ {∅}))
2119, 20mpbi 233 . . . . . . . . 9 (𝑤𝑥 ↦ (𝑦𝑤)):𝑥⟶(ran 𝑦 ∪ {∅})
22 vex 3483 . . . . . . . . 9 𝑥 ∈ V
23 vex 3483 . . . . . . . . . . 11 𝑦 ∈ V
2423rnex 7612 . . . . . . . . . 10 ran 𝑦 ∈ V
25 p0ex 5272 . . . . . . . . . 10 {∅} ∈ V
2624, 25unex 7463 . . . . . . . . 9 (ran 𝑦 ∪ {∅}) ∈ V
27 fex2 7633 . . . . . . . . 9 (((𝑤𝑥 ↦ (𝑦𝑤)):𝑥⟶(ran 𝑦 ∪ {∅}) ∧ 𝑥 ∈ V ∧ (ran 𝑦 ∪ {∅}) ∈ V) → (𝑤𝑥 ↦ (𝑦𝑤)) ∈ V)
2821, 22, 26, 27mp3an 1458 . . . . . . . 8 (𝑤𝑥 ↦ (𝑦𝑤)) ∈ V
29 fneq1 6432 . . . . . . . . 9 (𝑓 = (𝑤𝑥 ↦ (𝑦𝑤)) → (𝑓 Fn 𝑥 ↔ (𝑤𝑥 ↦ (𝑦𝑤)) Fn 𝑥))
30 fveq1 6660 . . . . . . . . . . . 12 (𝑓 = (𝑤𝑥 ↦ (𝑦𝑤)) → (𝑓𝑧) = ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧))
3130eleq1d 2900 . . . . . . . . . . 11 (𝑓 = (𝑤𝑥 ↦ (𝑦𝑤)) → ((𝑓𝑧) ∈ 𝑧 ↔ ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧))
3231imbi2d 344 . . . . . . . . . 10 (𝑓 = (𝑤𝑥 ↦ (𝑦𝑤)) → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧)))
3332ralbidv 3192 . . . . . . . . 9 (𝑓 = (𝑤𝑥 ↦ (𝑦𝑤)) → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧)))
3429, 33anbi12d 633 . . . . . . . 8 (𝑓 = (𝑤𝑥 ↦ (𝑦𝑤)) → ((𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) ↔ ((𝑤𝑥 ↦ (𝑦𝑤)) Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧))))
3528, 34spcev 3593 . . . . . . 7 (((𝑤𝑥 ↦ (𝑦𝑤)) Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧)) → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
3617, 35sylbir 238 . . . . . 6 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧) → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
3736exlimiv 1932 . . . . 5 (∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧) → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
386, 37sylbi 220 . . . 4 (∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
39 exsimpr 1871 . . . 4 (∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
4038, 39impbii 212 . . 3 (∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
4140albii 1821 . 2 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑥𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
421, 41bitri 278 1 (CHOICE ↔ ∀𝑥𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wex 1781  wcel 2115  wne 3014  wral 3133  Vcvv 3480  cun 3917  c0 4276  {csn 4550  cmpt 5132  ran crn 5543   Fn wfn 6338  wf 6339  cfv 6343  CHOICEwac 9539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ac 9540
This theorem is referenced by:  dfac5  9552  dfacacn  9565  ac5  9897
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