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Theorem dfac4 8983
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 8982 . 2 (CHOICE ↔ ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
2 fveq1 6228 . . . . . . . . 9 (𝑓 = 𝑦 → (𝑓𝑧) = (𝑦𝑧))
32eleq1d 2715 . . . . . . . 8 (𝑓 = 𝑦 → ((𝑓𝑧) ∈ 𝑧 ↔ (𝑦𝑧) ∈ 𝑧))
43imbi2d 329 . . . . . . 7 (𝑓 = 𝑦 → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧)))
54ralbidv 3015 . . . . . 6 (𝑓 = 𝑦 → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧)))
65cbvexv 2311 . . . . 5 (∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧))
7 fvex 6239 . . . . . . . . 9 (𝑦𝑤) ∈ V
8 eqid 2651 . . . . . . . . 9 (𝑤𝑥 ↦ (𝑦𝑤)) = (𝑤𝑥 ↦ (𝑦𝑤))
97, 8fnmpti 6060 . . . . . . . 8 (𝑤𝑥 ↦ (𝑦𝑤)) Fn 𝑥
10 fveq2 6229 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (𝑦𝑤) = (𝑦𝑧))
11 fvex 6239 . . . . . . . . . . . . 13 (𝑦𝑧) ∈ V
1210, 8, 11fvmpt 6321 . . . . . . . . . . . 12 (𝑧𝑥 → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) = (𝑦𝑧))
1312eleq1d 2715 . . . . . . . . . . 11 (𝑧𝑥 → (((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧 ↔ (𝑦𝑧) ∈ 𝑧))
1413imbi2d 329 . . . . . . . . . 10 (𝑧𝑥 → ((𝑧 ≠ ∅ → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧)))
1514ralbiia 3008 . . . . . . . . 9 (∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧))
1615anbi2i 730 . . . . . . . 8 (((𝑤𝑥 ↦ (𝑦𝑤)) Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧)) ↔ ((𝑤𝑥 ↦ (𝑦𝑤)) Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧)))
179, 16mpbiran 973 . . . . . . 7 (((𝑤𝑥 ↦ (𝑦𝑤)) Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧)) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧))
18 fvrn0 6254 . . . . . . . . . . 11 (𝑦𝑤) ∈ (ran 𝑦 ∪ {∅})
1918rgenw 2953 . . . . . . . . . 10 𝑤𝑥 (𝑦𝑤) ∈ (ran 𝑦 ∪ {∅})
208fmpt 6421 . . . . . . . . . 10 (∀𝑤𝑥 (𝑦𝑤) ∈ (ran 𝑦 ∪ {∅}) ↔ (𝑤𝑥 ↦ (𝑦𝑤)):𝑥⟶(ran 𝑦 ∪ {∅}))
2119, 20mpbi 220 . . . . . . . . 9 (𝑤𝑥 ↦ (𝑦𝑤)):𝑥⟶(ran 𝑦 ∪ {∅})
22 vex 3234 . . . . . . . . 9 𝑥 ∈ V
23 vex 3234 . . . . . . . . . . 11 𝑦 ∈ V
2423rnex 7142 . . . . . . . . . 10 ran 𝑦 ∈ V
25 p0ex 4883 . . . . . . . . . 10 {∅} ∈ V
2624, 25unex 6998 . . . . . . . . 9 (ran 𝑦 ∪ {∅}) ∈ V
27 fex2 7163 . . . . . . . . 9 (((𝑤𝑥 ↦ (𝑦𝑤)):𝑥⟶(ran 𝑦 ∪ {∅}) ∧ 𝑥 ∈ V ∧ (ran 𝑦 ∪ {∅}) ∈ V) → (𝑤𝑥 ↦ (𝑦𝑤)) ∈ V)
2821, 22, 26, 27mp3an 1464 . . . . . . . 8 (𝑤𝑥 ↦ (𝑦𝑤)) ∈ V
29 fneq1 6017 . . . . . . . . 9 (𝑓 = (𝑤𝑥 ↦ (𝑦𝑤)) → (𝑓 Fn 𝑥 ↔ (𝑤𝑥 ↦ (𝑦𝑤)) Fn 𝑥))
30 fveq1 6228 . . . . . . . . . . . 12 (𝑓 = (𝑤𝑥 ↦ (𝑦𝑤)) → (𝑓𝑧) = ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧))
3130eleq1d 2715 . . . . . . . . . . 11 (𝑓 = (𝑤𝑥 ↦ (𝑦𝑤)) → ((𝑓𝑧) ∈ 𝑧 ↔ ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧))
3231imbi2d 329 . . . . . . . . . 10 (𝑓 = (𝑤𝑥 ↦ (𝑦𝑤)) → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧)))
3332ralbidv 3015 . . . . . . . . 9 (𝑓 = (𝑤𝑥 ↦ (𝑦𝑤)) → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧)))
3429, 33anbi12d 747 . . . . . . . 8 (𝑓 = (𝑤𝑥 ↦ (𝑦𝑤)) → ((𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) ↔ ((𝑤𝑥 ↦ (𝑦𝑤)) Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧))))
3528, 34spcev 3331 . . . . . . 7 (((𝑤𝑥 ↦ (𝑦𝑤)) Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧)) → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
3617, 35sylbir 225 . . . . . 6 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧) → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
3736exlimiv 1898 . . . . 5 (∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧) → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
386, 37sylbi 207 . . . 4 (∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
39 exsimpr 1836 . . . 4 (∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
4038, 39impbii 199 . . 3 (∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
4140albii 1787 . 2 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑥𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
421, 41bitri 264 1 (CHOICE ↔ ∀𝑥𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
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  cun 3605  c0 3948  {csn 4210  cmpt 4762  ran crn 5144   Fn wfn 5921  wf 5922  cfv 5926  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-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-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-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  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-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ac 8977
This theorem is referenced by:  dfac5  8989  dfacacn  9001  ac5  9337
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