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Theorem opelresi 5856
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 5854 . 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 208  wa 398  wcel 2110  Vcvv 3495  cop 4567  cres 5552
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-opab 5122  df-xp 5556  df-res 5562
This theorem is referenced by:  opres  5858  dmres  5870  relssres  5888  iss  5898  restidsing  5917  asymref  5971  ssrnres  6030  cnvresima  6082  ressn  6131  funssres  6393  fcnvres  6551  fvn0ssdmfun  6837  relexpindlem  14416  dprd2dlem1  19157  dprd2da  19158  hausdiag  22247  hauseqlcld  22248  ovoliunlem1  24097  undmrnresiss  39957
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