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Theorem opelresi 5988
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 5986 . 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 205  wa 394  wcel 2104  Vcvv 3472  cop 4633  cres 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-opab 5210  df-xp 5681  df-res 5687
This theorem is referenced by:  opres  5990  dmres  6002  relssres  6021  iss  6034  restidsing  6051  asymref  6116  ssrnres  6176  cnvresima  6228  ressn  6283  funssres  6591  fcnvres  6767  fvn0ssdmfun  7075  relexpindlem  15014  dprd2dlem1  19952  dprd2da  19953  hausdiag  23369  hauseqlcld  23370  ovoliunlem1  25251  undmrnresiss  42657
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