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Theorem opelresi 5877
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 5875 . 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 209  wa 399  wcel 2112  Vcvv 3423  cop 4564  cres 5571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2710  ax-sep 5209  ax-nul 5216  ax-pr 5339
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3425  df-dif 3886  df-un 3888  df-in 3890  df-nul 4255  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-opab 5133  df-xp 5575  df-res 5581
This theorem is referenced by:  opres  5879  dmres  5891  relssres  5910  iss  5921  restidsing  5940  asymref  5999  ssrnres  6059  cnvresima  6111  ressn  6166  funssres  6445  fcnvres  6618  fvn0ssdmfun  6917  relexpindlem  14659  dprd2dlem1  19461  dprd2da  19462  hausdiag  22574  hauseqlcld  22575  ovoliunlem1  24431  undmrnresiss  40936
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