MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfac2a Structured version   Visualization version   GIF version

Theorem dfac2a 9173
Description: Our Axiom of Choice (in the form of ac3 9507) implies the Axiom of Choice (first form) of [Enderton] p. 49. The proof uses neither AC nor the Axiom of Regularity. See dfac2b 9174 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 6779 . . . . . . . . 9 (∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣) → (𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) = {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)})
2 riotacl 6787 . . . . . . . . 9 (∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣) → (𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) ∈ 𝑧)
31, 2eqeltrrd 2854 . . . . . . . 8 (∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣) → {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)} ∈ 𝑧)
4 elequ2 2162 . . . . . . . . . . . . 13 (𝑢 = 𝑧 → (𝑤𝑢𝑤𝑧))
5 elequ1 2155 . . . . . . . . . . . . . . 15 (𝑢 = 𝑧 → (𝑢𝑣𝑧𝑣))
65anbi1d 616 . . . . . . . . . . . . . 14 (𝑢 = 𝑧 → ((𝑢𝑣𝑤𝑣) ↔ (𝑧𝑣𝑤𝑣)))
76rexbidv 3204 . . . . . . . . . . . . 13 (𝑢 = 𝑧 → (∃𝑣𝑦 (𝑢𝑣𝑤𝑣) ↔ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)))
84, 7anbi12d 617 . . . . . . . . . . . 12 (𝑢 = 𝑧 → ((𝑤𝑢 ∧ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)) ↔ (𝑤𝑧 ∧ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣))))
98rabbidva2 3340 . . . . . . . . . . 11 (𝑢 = 𝑧 → {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} = {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)})
109unieqd 4595 . . . . . . . . . 10 (𝑢 = 𝑧 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} = {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)})
11 eqid 2774 . . . . . . . . . 10 (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}) = (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})
12 vex 3358 . . . . . . . . . . . 12 𝑧 ∈ V
1312rabex 4960 . . . . . . . . . . 11 {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)} ∈ V
1413uniex 7121 . . . . . . . . . 10 {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)} ∈ V
1510, 11, 14fvmpt 6441 . . . . . . . . 9 (𝑧𝑥 → ((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})‘𝑧) = {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)})
1615eleq1d 2838 . . . . . . . 8 (𝑧𝑥 → (((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})‘𝑧) ∈ 𝑧 {𝑤𝑧 ∣ ∃𝑣𝑦 (𝑧𝑣𝑤𝑣)} ∈ 𝑧))
173, 16syl5ibr 237 . . . . . . 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 3102 . . . . 5 (∀𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) → ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})‘𝑧) ∈ 𝑧))
20 ssrab2 3843 . . . . . . . . . . 11 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} ⊆ 𝑢
21 elssuni 4614 . . . . . . . . . . 11 (𝑢𝑥𝑢 𝑥)
2220, 21syl5ss 3769 . . . . . . . . . 10 (𝑢𝑥 → {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} ⊆ 𝑥)
2322unissd 4609 . . . . . . . . 9 (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} ⊆ 𝑥)
24 vex 3358 . . . . . . . . . . . 12 𝑥 ∈ V
2524uniex 7121 . . . . . . . . . . 11 𝑥 ∈ V
2625uniex 7121 . . . . . . . . . 10 𝑥 ∈ V
2726elpw2 4973 . . . . . . . . 9 ( {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} ∈ 𝒫 𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} ⊆ 𝑥)
2823, 27sylibr 225 . . . . . . . 8 (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)} ∈ 𝒫 𝑥)
2911, 28fmpti 6542 . . . . . . 7 (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}):𝑥⟶𝒫 𝑥
3026pwex 4995 . . . . . . 7 𝒫 𝑥 ∈ V
31 fex2 7289 . . . . . . 7 (((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}):𝑥⟶𝒫 𝑥𝑥 ∈ V ∧ 𝒫 𝑥 ∈ V) → (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}) ∈ V)
3229, 24, 30, 31mp3an 1575 . . . . . 6 (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}) ∈ V
33 fveq1 6347 . . . . . . . . 9 (𝑓 = (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}) → (𝑓𝑧) = ((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})‘𝑧))
3433eleq1d 2838 . . . . . . . 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 3138 . . . . . 6 (𝑓 = (𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)}) → (∀𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧) ↔ ∀𝑧𝑥 (𝑧 ≠ ∅ → ((𝑢𝑥 {𝑤𝑢 ∣ ∃𝑣𝑦 (𝑢𝑣𝑤𝑣)})‘𝑧) ∈ 𝑧)))
3732, 36spcev 3456 . . . . 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 2013 . . 3 (∃𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) → ∃𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
4039alimi 1890 . 2 (∀𝑥𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) → ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
41 dfac3 9165 . 2 (CHOICE ↔ ∀𝑥𝑓𝑧𝑥 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧))
4240, 41sylibr 225 1 (∀𝑥𝑦𝑧𝑥 (𝑧 ≠ ∅ → ∃!𝑤𝑧𝑣𝑦 (𝑧𝑣𝑤𝑣)) → CHOICE)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1632   = wceq 1634  wex 1855  wcel 2148  wne 2946  wral 3064  wrex 3065  ∃!wreu 3066  {crab 3068  Vcvv 3355  wss 3729  c0 4073  𝒫 cpw 4307   cuni 4585  cmpt 4876  wf 6038  cfv 6042  crio 6772  CHOICEwac 9159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3357  df-sbc 3594  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-op 4333  df-uni 4586  df-br 4798  df-opab 4860  df-mpt 4877  df-id 5171  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-fv 6050  df-riota 6773  df-ac 9160
This theorem is referenced by:  dfac2  9175  dfac2OLD  9176  axac2  9511
  Copyright terms: Public domain W3C validator