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Theorem bj-opelresdm 37128
Description: If an ordered pair is in a restricted binary relation, then its first component is an element of the restricting class. See also opelres 6006. (Contributed by BJ, 25-Dec-2023.)
Assertion
Ref Expression
bj-opelresdm (⟨𝐴, 𝐵⟩ ∈ (𝑅𝑋) → 𝐴𝑋)

Proof of Theorem bj-opelresdm
StepHypRef Expression
1 elin 3979 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ (𝑋 × V)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × V)))
2 opelxp1 5731 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝑋 × V) → 𝐴𝑋)
31, 2simplbiim 504 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ (𝑋 × V)) → 𝐴𝑋)
4 df-res 5701 . 2 (𝑅𝑋) = (𝑅 ∩ (𝑋 × V))
53, 4eleq2s 2857 1 (⟨𝐴, 𝐵⟩ ∈ (𝑅𝑋) → 𝐴𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  Vcvv 3478  cin 3962  cop 4637   × cxp 5687  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:  bj-brresdm  37129
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