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Theorem opres 6007
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 6005 . 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 2108  Vcvv 3480  cop 4632  cres 5687
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-opab 5206  df-xp 5691  df-res 5697
This theorem is referenced by:  resieq  6008  2elresin  6689  mdetunilem9  22626
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