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

Theorem opabex 7213
Description: Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.)
Hypotheses
Ref Expression
opabex.1 𝐴 ∈ V
opabex.2 (𝑥𝐴 → ∃*𝑦𝜑)
Assertion
Ref Expression
opabex {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem opabex
StepHypRef Expression
1 funopab 6573 . . 3 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥∃*𝑦(𝑥𝐴𝜑))
2 opabex.2 . . . 4 (𝑥𝐴 → ∃*𝑦𝜑)
3 moanimv 2607 . . . 4 (∃*𝑦(𝑥𝐴𝜑) ↔ (𝑥𝐴 → ∃*𝑦𝜑))
42, 3mpbir 230 . . 3 ∃*𝑦(𝑥𝐴𝜑)
51, 4mpgbir 1793 . 2 Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
6 opabex.1 . . 3 𝐴 ∈ V
7 dmopabss 5908 . . 3 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
86, 7ssexi 5312 . 2 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V
9 funex 7212 . 2 ((Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V)
105, 8, 9mp2an 689 1 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2098  ∃*wmo 2524  Vcvv 3466  {copab 5200  dom cdm 5666  Fun wfun 6527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator