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Theorem dfac4 8828
 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 8827 . 2 (CHOICE ↔ ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
2 fveq1 6102 . . . . . . . . 9 (𝑓 = 𝑦 → (𝑓𝑧) = (𝑦𝑧))
32eleq1d 2672 . . . . . . . 8 (𝑓 = 𝑦 → ((𝑓𝑧) ∈ 𝑧 ↔ (𝑦𝑧) ∈ 𝑧))
43imbi2d 329 . . . . . . 7 (𝑓 = 𝑦 → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧)))
54ralbidv 2969 . . . . . 6 (𝑓 = 𝑦 → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧)))
65cbvexv 2263 . . . . 5 (∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧))
7 fvex 6113 . . . . . . . . 9 (𝑦𝑤) ∈ V
8 eqid 2610 . . . . . . . . 9 (𝑤𝑥 ↦ (𝑦𝑤)) = (𝑤𝑥 ↦ (𝑦𝑤))
97, 8fnmpti 5935 . . . . . . . 8 (𝑤𝑥 ↦ (𝑦𝑤)) Fn 𝑥
10 fveq2 6103 . . . . . . . . . . . . 13 (𝑤 = 𝑧 → (𝑦𝑤) = (𝑦𝑧))
11 fvex 6113 . . . . . . . . . . . . 13 (𝑦𝑧) ∈ V
1210, 8, 11fvmpt 6191 . . . . . . . . . . . 12 (𝑧𝑥 → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) = (𝑦𝑧))
1312eleq1d 2672 . . . . . . . . . . 11 (𝑧𝑥 → (((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧 ↔ (𝑦𝑧) ∈ 𝑧))
1413imbi2d 329 . . . . . . . . . 10 (𝑧𝑥 → ((𝑧 ≠ ∅ → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧)))
1514ralbiia 2962 . . . . . . . . 9 (∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧))
1615anbi2i 726 . . . . . . . 8 (((𝑤𝑥 ↦ (𝑦𝑤)) Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧)) ↔ ((𝑤𝑥 ↦ (𝑦𝑤)) Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧)))
179, 16mpbiran 955 . . . . . . 7 (((𝑤𝑥 ↦ (𝑦𝑤)) Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧)) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧))
18 fvrn0 6126 . . . . . . . . . . 11 (𝑦𝑤) ∈ (ran 𝑦 ∪ {∅})
1918rgenw 2908 . . . . . . . . . 10 𝑤𝑥 (𝑦𝑤) ∈ (ran 𝑦 ∪ {∅})
208fmpt 6289 . . . . . . . . . 10 (∀𝑤𝑥 (𝑦𝑤) ∈ (ran 𝑦 ∪ {∅}) ↔ (𝑤𝑥 ↦ (𝑦𝑤)):𝑥⟶(ran 𝑦 ∪ {∅}))
2119, 20mpbi 219 . . . . . . . . 9 (𝑤𝑥 ↦ (𝑦𝑤)):𝑥⟶(ran 𝑦 ∪ {∅})
22 vex 3176 . . . . . . . . 9 𝑥 ∈ V
23 vex 3176 . . . . . . . . . . 11 𝑦 ∈ V
2423rnex 6992 . . . . . . . . . 10 ran 𝑦 ∈ V
25 p0ex 4779 . . . . . . . . . 10 {∅} ∈ V
2624, 25unex 6854 . . . . . . . . 9 (ran 𝑦 ∪ {∅}) ∈ V
27 fex2 7014 . . . . . . . . 9 (((𝑤𝑥 ↦ (𝑦𝑤)):𝑥⟶(ran 𝑦 ∪ {∅}) ∧ 𝑥 ∈ V ∧ (ran 𝑦 ∪ {∅}) ∈ V) → (𝑤𝑥 ↦ (𝑦𝑤)) ∈ V)
2821, 22, 26, 27mp3an 1416 . . . . . . . 8 (𝑤𝑥 ↦ (𝑦𝑤)) ∈ V
29 fneq1 5893 . . . . . . . . 9 (𝑓 = (𝑤𝑥 ↦ (𝑦𝑤)) → (𝑓 Fn 𝑥 ↔ (𝑤𝑥 ↦ (𝑦𝑤)) Fn 𝑥))
30 fveq1 6102 . . . . . . . . . . . 12 (𝑓 = (𝑤𝑥 ↦ (𝑦𝑤)) → (𝑓𝑧) = ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧))
3130eleq1d 2672 . . . . . . . . . . 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 2969 . . . . . . . . 9 (𝑓 = (𝑤𝑥 ↦ (𝑦𝑤)) → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧)))
3429, 33anbi12d 743 . . . . . . . 8 (𝑓 = (𝑤𝑥 ↦ (𝑦𝑤)) → ((𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) ↔ ((𝑤𝑥 ↦ (𝑦𝑤)) Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧))))
3528, 34spcev 3273 . . . . . . 7 (((𝑤𝑥 ↦ (𝑦𝑤)) Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑤𝑥 ↦ (𝑦𝑤))‘𝑧) ∈ 𝑧)) → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
3617, 35sylbir 224 . . . . . 6 (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧) → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
3736exlimiv 1845 . . . . 5 (∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → (𝑦𝑧) ∈ 𝑧) → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
386, 37sylbi 206 . . . 4 (∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
39 exsimpr 1784 . . . 4 (∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)) → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
4038, 39impbii 198 . . 3 (∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
4140albii 1737 . 2 (∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑥𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
421, 41bitri 263 1 (CHOICE ↔ ∀𝑥𝑓(𝑓 Fn 𝑥 ∧ ∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∪ cun 3538  ∅c0 3874  {csn 4125   ↦ cmpt 4643  ran crn 5039   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  CHOICEwac 8821 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ac 8822 This theorem is referenced by:  dfac5  8834  dfacacn  8846  ac5  9182
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