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

Theorem opabex 6840
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 6252 . . 3 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥∃*𝑦(𝑥𝐴𝜑))
2 opabex.2 . . . 4 (𝑥𝐴 → ∃*𝑦𝜑)
3 moanimv 2670 . . . 4 (∃*𝑦(𝑥𝐴𝜑) ↔ (𝑥𝐴 → ∃*𝑦𝜑))
42, 3mpbir 232 . . 3 ∃*𝑦(𝑥𝐴𝜑)
51, 4mpgbir 1779 . 2 Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}
6 opabex.1 . . 3 𝐴 ∈ V
7 dmopabss 5665 . . 3 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
86, 7ssexi 5110 . 2 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V
9 funex 6839 . 2 ((Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V)
105, 8, 9mp2an 688 1 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2079  ∃*wmo 2572  Vcvv 3432  {copab 5018  dom cdm 5435  Fun wfun 6211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pr 5214
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator