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Theorem dfac2a 8896
Description: Our Axiom of Choice (in the form of ac3 9228) implies the Axiom of Choice (first form) of [Enderton] p. 49. The proof uses neither AC nor the Axiom of Regularity. See dfac2 8897 for the converse (which does use the Axiom of Regularity). (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 26-Jun-2015.)
Assertion
Ref Expression
dfac2a (∀𝑥𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) → CHOICE)
Distinct variable group:   𝑥,𝑧,𝑦,𝑤,𝑣

Proof of Theorem dfac2a
Dummy variables 𝑓 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 riotauni 6571 . . . . . . . . 9 (∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣) → (𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) = {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)})
2 riotacl 6579 . . . . . . . . 9 (∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣) → (𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) ∈ 𝑧)
31, 2eqeltrrd 2699 . . . . . . . 8 (∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣) → {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)} ∈ 𝑧)
4 elequ2 2001 . . . . . . . . . . . . 13 (𝑢 = 𝑧 → (𝑤𝑢𝑤𝑧))
5 elequ1 1994 . . . . . . . . . . . . . . 15 (𝑢 = 𝑧 → (𝑢𝑣𝑧𝑣))
65anbi1d 740 . . . . . . . . . . . . . 14 (𝑢 = 𝑧 → ((𝑢𝑣𝑤𝑣) ↔ (𝑧𝑣𝑤𝑣)))
76rexbidv 3045 . . . . . . . . . . . . 13 (𝑢 = 𝑧 → (∃𝑣𝑦 (𝑢𝑣𝑤𝑣) ↔ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)))
84, 7anbi12d 746 . . . . . . . . . . . 12 (𝑢 = 𝑧 → ((𝑤𝑢 ∧ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)) ↔ (𝑤𝑧 ∧ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣))))
98rabbidva2 3174 . . . . . . . . . . 11 (𝑢 = 𝑧 → {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} = {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)})
109unieqd 4412 . . . . . . . . . 10 (𝑢 = 𝑧 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} = {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)})
11 eqid 2621 . . . . . . . . . 10 (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}) = (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})
12 vex 3189 . . . . . . . . . . . 12 𝑧 ∈ V
1312rabex 4773 . . . . . . . . . . 11 {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)} ∈ V
1413uniex 6906 . . . . . . . . . 10 {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)} ∈ V
1510, 11, 14fvmpt 6239 . . . . . . . . 9 (𝑧𝑥 → ((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})‘𝑧) = {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)})
1615eleq1d 2683 . . . . . . . 8 (𝑧𝑥 → (((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})‘𝑧) ∈ 𝑧 {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)} ∈ 𝑧))
173, 16syl5ibr 236 . . . . . . 7 (𝑧𝑥 → (∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣) → ((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})‘𝑧) ∈ 𝑧))
1817imim2d 57 . . . . . 6 (𝑧𝑥 → ((𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) → (𝑧 ≠ ∅ → ((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})‘𝑧) ∈ 𝑧)))
1918ralimia 2945 . . . . 5 (∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) → ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})‘𝑧) ∈ 𝑧))
20 ssrab2 3666 . . . . . . . . . . 11 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} ⊆ 𝑢
21 elssuni 4433 . . . . . . . . . . 11 (𝑢𝑥𝑢 𝑥)
2220, 21syl5ss 3594 . . . . . . . . . 10 (𝑢𝑥 → {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} ⊆ 𝑥)
2322unissd 4428 . . . . . . . . 9 (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} ⊆ 𝑥)
24 vex 3189 . . . . . . . . . . . 12 𝑥 ∈ V
2524uniex 6906 . . . . . . . . . . 11 𝑥 ∈ V
2625uniex 6906 . . . . . . . . . 10 𝑥 ∈ V
2726elpw2 4788 . . . . . . . . 9 ( {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} ∈ 𝒫 𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} ⊆ 𝑥)
2823, 27sylibr 224 . . . . . . . 8 (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} ∈ 𝒫 𝑥)
2911, 28fmpti 6339 . . . . . . 7 (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}):𝑥⟶𝒫 𝑥
3026pwex 4808 . . . . . . 7 𝒫 𝑥 ∈ V
31 fex2 7068 . . . . . . 7 (((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}):𝑥⟶𝒫 𝑥𝑥 ∈ V ∧ 𝒫 𝑥 ∈ V) → (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}) ∈ V)
3229, 24, 30, 31mp3an 1421 . . . . . 6 (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}) ∈ V
33 fveq1 6147 . . . . . . . . 9 (𝑓 = (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}) → (𝑓𝑧) = ((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})‘𝑧))
3433eleq1d 2683 . . . . . . . 8 (𝑓 = (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}) → ((𝑓𝑧) ∈ 𝑧 ↔ ((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})‘𝑧) ∈ 𝑧))
3534imbi2d 330 . . . . . . 7 (𝑓 = (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}) → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → ((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})‘𝑧) ∈ 𝑧)))
3635ralbidv 2980 . . . . . 6 (𝑓 = (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}) → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})‘𝑧) ∈ 𝑧)))
3732, 36spcev 3286 . . . . 5 (∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})‘𝑧) ∈ 𝑧) → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
3819, 37syl 17 . . . 4 (∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
3938exlimiv 1855 . . 3 (∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
4039alimi 1736 . 2 (∀𝑥𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) → ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
41 dfac3 8888 . 2 (CHOICE ↔ ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
4240, 41sylibr 224 1 (∀𝑥𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) → CHOICE)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  wrex 2908  ∃!wreu 2909  {crab 2911  Vcvv 3186  wss 3555  c0 3891  𝒫 cpw 4130   cuni 4402  cmpt 4673  wf 5843  cfv 5847  crio 6564  CHOICEwac 8882
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-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-riota 6565  df-ac 8883
This theorem is referenced by:  dfac2  8897  axac2  9232
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