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Theorem dfac2a 9140
 Description: Our Axiom of Choice (in the form of ac3 9474) implies the Axiom of Choice (first form) of [Enderton] p. 49. The proof uses neither AC nor the Axiom of Regularity. See dfac2b 9141 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 6778 . . . . . . . . 9 (∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣) → (𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) = {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)})
2 riotacl 6786 . . . . . . . . 9 (∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣) → (𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) ∈ 𝑧)
31, 2eqeltrrd 2838 . . . . . . . 8 (∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣) → {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)} ∈ 𝑧)
4 elequ2 2151 . . . . . . . . . . . . 13 (𝑢 = 𝑧 → (𝑤𝑢𝑤𝑧))
5 elequ1 2144 . . . . . . . . . . . . . . 15 (𝑢 = 𝑧 → (𝑢𝑣𝑧𝑣))
65anbi1d 743 . . . . . . . . . . . . . 14 (𝑢 = 𝑧 → ((𝑢𝑣𝑤𝑣) ↔ (𝑧𝑣𝑤𝑣)))
76rexbidv 3188 . . . . . . . . . . . . 13 (𝑢 = 𝑧 → (∃𝑣𝑦 (𝑢𝑣𝑤𝑣) ↔ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)))
84, 7anbi12d 749 . . . . . . . . . . . 12 (𝑢 = 𝑧 → ((𝑤𝑢 ∧ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)) ↔ (𝑤𝑧 ∧ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣))))
98rabbidva2 3324 . . . . . . . . . . 11 (𝑢 = 𝑧 → {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} = {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)})
109unieqd 4596 . . . . . . . . . 10 (𝑢 = 𝑧 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} = {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)})
11 eqid 2758 . . . . . . . . . 10 (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}) = (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})
12 vex 3341 . . . . . . . . . . . 12 𝑧 ∈ V
1312rabex 4962 . . . . . . . . . . 11 {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)} ∈ V
1413uniex 7116 . . . . . . . . . 10 {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)} ∈ V
1510, 11, 14fvmpt 6442 . . . . . . . . 9 (𝑧𝑥 → ((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})‘𝑧) = {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)})
1615eleq1d 2822 . . . . . . . 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 3086 . . . . 5 (∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) → ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})‘𝑧) ∈ 𝑧))
20 ssrab2 3826 . . . . . . . . . . 11 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} ⊆ 𝑢
21 elssuni 4617 . . . . . . . . . . 11 (𝑢𝑥𝑢 𝑥)
2220, 21syl5ss 3753 . . . . . . . . . 10 (𝑢𝑥 → {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} ⊆ 𝑥)
2322unissd 4612 . . . . . . . . 9 (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} ⊆ 𝑥)
24 vex 3341 . . . . . . . . . . . 12 𝑥 ∈ V
2524uniex 7116 . . . . . . . . . . 11 𝑥 ∈ V
2625uniex 7116 . . . . . . . . . 10 𝑥 ∈ V
2726elpw2 4975 . . . . . . . . 9 ( {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} ∈ 𝒫 𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} ⊆ 𝑥)
2823, 27sylibr 224 . . . . . . . 8 (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} ∈ 𝒫 𝑥)
2911, 28fmpti 6544 . . . . . . 7 (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}):𝑥⟶𝒫 𝑥
3026pwex 4995 . . . . . . 7 𝒫 𝑥 ∈ V
31 fex2 7284 . . . . . . 7 (((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}):𝑥⟶𝒫 𝑥𝑥 ∈ V ∧ 𝒫 𝑥 ∈ V) → (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}) ∈ V)
3229, 24, 30, 31mp3an 1571 . . . . . 6 (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}) ∈ V
33 fveq1 6349 . . . . . . . . 9 (𝑓 = (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}) → (𝑓𝑧) = ((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})‘𝑧))
3433eleq1d 2822 . . . . . . . 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 3122 . . . . . 6 (𝑓 = (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}) → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})‘𝑧) ∈ 𝑧)))
3732, 36spcev 3438 . . . . 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 2005 . . 3 (∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
4039alimi 1886 . 2 (∀𝑥𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) → ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
41 dfac3 9132 . 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 1628   = wceq 1630  ∃wex 1851   ∈ wcel 2137   ≠ wne 2930  ∀wral 3048  ∃wrex 3049  ∃!wreu 3050  {crab 3052  Vcvv 3338   ⊆ wss 3713  ∅c0 4056  𝒫 cpw 4300  ∪ cuni 4586   ↦ cmpt 4879  ⟶wf 6043  ‘cfv 6047  ℩crio 6771  CHOICEwac 9126 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-fv 6055  df-riota 6772  df-ac 9127 This theorem is referenced by:  dfac2  9142  dfac2OLD  9143  axac2  9478
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