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Theorem opres 6010
Description: Ordered pair membership in a restriction when the first member belongs to the restricting class. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Hypothesis
Ref Expression
opres.1 𝐵 ∈ V
Assertion
Ref Expression
opres (𝐴𝐷 → (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))

Proof of Theorem opres
StepHypRef Expression
1 opres.1 . . 3 𝐵 ∈ V
21opelresi 6008 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ (𝐴𝐷 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶))
32baib 535 1 (𝐴𝐷 → (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  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:  resieq  6011  2elresin  6690  mdetunilem9  22642
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