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Theorem dfac2a 8990
Description: Our Axiom of Choice (in the form of ac3 9322) implies the Axiom of Choice (first form) of [Enderton] p. 49. The proof uses neither AC nor the Axiom of Regularity. See dfac2 8991 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 6657 . . . . . . . . 9 (∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣) → (𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) = {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)})
2 riotacl 6665 . . . . . . . . 9 (∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣) → (𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) ∈ 𝑧)
31, 2eqeltrrd 2731 . . . . . . . 8 (∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣) → {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)} ∈ 𝑧)
4 elequ2 2044 . . . . . . . . . . . . 13 (𝑢 = 𝑧 → (𝑤𝑢𝑤𝑧))
5 elequ1 2037 . . . . . . . . . . . . . . 15 (𝑢 = 𝑧 → (𝑢𝑣𝑧𝑣))
65anbi1d 741 . . . . . . . . . . . . . 14 (𝑢 = 𝑧 → ((𝑢𝑣𝑤𝑣) ↔ (𝑧𝑣𝑤𝑣)))
76rexbidv 3081 . . . . . . . . . . . . 13 (𝑢 = 𝑧 → (∃𝑣𝑦 (𝑢𝑣𝑤𝑣) ↔ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)))
84, 7anbi12d 747 . . . . . . . . . . . 12 (𝑢 = 𝑧 → ((𝑤𝑢 ∧ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)) ↔ (𝑤𝑧 ∧ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣))))
98rabbidva2 3217 . . . . . . . . . . 11 (𝑢 = 𝑧 → {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} = {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)})
109unieqd 4478 . . . . . . . . . 10 (𝑢 = 𝑧 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} = {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)})
11 eqid 2651 . . . . . . . . . 10 (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}) = (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})
12 vex 3234 . . . . . . . . . . . 12 𝑧 ∈ V
1312rabex 4845 . . . . . . . . . . 11 {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)} ∈ V
1413uniex 6995 . . . . . . . . . 10 {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)} ∈ V
1510, 11, 14fvmpt 6321 . . . . . . . . 9 (𝑧𝑥 → ((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})‘𝑧) = {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)})
1615eleq1d 2715 . . . . . . . 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 2979 . . . . 5 (∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) → ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})‘𝑧) ∈ 𝑧))
20 ssrab2 3720 . . . . . . . . . . 11 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} ⊆ 𝑢
21 elssuni 4499 . . . . . . . . . . 11 (𝑢𝑥𝑢 𝑥)
2220, 21syl5ss 3647 . . . . . . . . . 10 (𝑢𝑥 → {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} ⊆ 𝑥)
2322unissd 4494 . . . . . . . . 9 (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} ⊆ 𝑥)
24 vex 3234 . . . . . . . . . . . 12 𝑥 ∈ V
2524uniex 6995 . . . . . . . . . . 11 𝑥 ∈ V
2625uniex 6995 . . . . . . . . . 10 𝑥 ∈ V
2726elpw2 4858 . . . . . . . . 9 ( {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} ∈ 𝒫 𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} ⊆ 𝑥)
2823, 27sylibr 224 . . . . . . . 8 (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} ∈ 𝒫 𝑥)
2911, 28fmpti 6423 . . . . . . 7 (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}):𝑥⟶𝒫 𝑥
3026pwex 4878 . . . . . . 7 𝒫 𝑥 ∈ V
31 fex2 7163 . . . . . . 7 (((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}):𝑥⟶𝒫 𝑥𝑥 ∈ V ∧ 𝒫 𝑥 ∈ V) → (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}) ∈ V)
3229, 24, 30, 31mp3an 1464 . . . . . 6 (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}) ∈ V
33 fveq1 6228 . . . . . . . . 9 (𝑓 = (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}) → (𝑓𝑧) = ((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})‘𝑧))
3433eleq1d 2715 . . . . . . . 8 (𝑓 = (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}) → ((𝑓𝑧) ∈ 𝑧 ↔ ((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})‘𝑧) ∈ 𝑧))
3534imbi2d 329 . . . . . . 7 (𝑓 = (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}) → ((𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ (𝑧 ≠ ∅ → ((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})‘𝑧) ∈ 𝑧)))
3635ralbidv 3015 . . . . . 6 (𝑓 = (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}) → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})‘𝑧) ∈ 𝑧)))
3732, 36spcev 3331 . . . . 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 1898 . . 3 (∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
4039alimi 1779 . 2 (∀𝑥𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) → ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
41 dfac3 8982 . 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 383  wal 1521   = wceq 1523  wex 1744  wcel 2030  wne 2823  wral 2941  wrex 2942  ∃!wreu 2943  {crab 2945  Vcvv 3231  wss 3607  c0 3948  𝒫 cpw 4191   cuni 4468  cmpt 4762  wf 5922  cfv 5926  crio 6650  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-reu 2948  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-riota 6651  df-ac 8977
This theorem is referenced by:  dfac2  8991  axac2  9326
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