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Theorem opelresi 6008
Description: Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.)
Hypothesis
Ref Expression
opelresi.1 𝐶 ∈ V
Assertion
Ref Expression
opelresi (⟨𝐵, 𝐶⟩ ∈ (𝑅𝐴) ↔ (𝐵𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅))

Proof of Theorem opelresi
StepHypRef Expression
1 opelresi.1 . 2 𝐶 ∈ V
2 opelres 6006 . 2 (𝐶 ∈ V → (⟨𝐵, 𝐶⟩ ∈ (𝑅𝐴) ↔ (𝐵𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅)))
31, 2ax-mp 5 1 (⟨𝐵, 𝐶⟩ ∈ (𝑅𝐴) ↔ (𝐵𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2106  Vcvv 3478  cop 4637  cres 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-xp 5695  df-res 5701
This theorem is referenced by:  opres  6010  dmres  6032  relssres  6042  iss  6055  restidsing  6073  asymref  6139  ssrnres  6200  cnvresima  6252  ressn  6307  funssres  6612  fcnvres  6786  fvn0ssdmfun  7094  relexpindlem  15099  dprd2dlem1  20076  dprd2da  20077  hausdiag  23669  hauseqlcld  23670  ovoliunlem1  25551  undmrnresiss  43594
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