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Theorem opelresi 5941
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 5939 . 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 2111  Vcvv 3436  cop 4581  cres 5621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pr 5372
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-opab 5156  df-xp 5625  df-res 5631
This theorem is referenced by:  opres  5943  dmres  5966  relssres  5976  iss  5989  restidsing  6007  asymref  6068  ssrnres  6131  cnvresima  6183  ressn  6238  funssres  6531  fcnvres  6706  fvn0ssdmfun  7013  relexpindlem  14976  dprd2dlem1  19961  dprd2da  19962  hausdiag  23566  hauseqlcld  23567  ovoliunlem1  25436  undmrnresiss  43702
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