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Theorem bj-opelresdm 34959
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 5831. (Contributed by BJ, 25-Dec-2023.)
Assertion
Ref Expression
bj-opelresdm (⟨𝐴, 𝐵⟩ ∈ (𝑅𝑋) → 𝐴𝑋)

Proof of Theorem bj-opelresdm
StepHypRef Expression
1 elin 3859 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ (𝑋 × V)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × V)))
2 opelxp1 5566 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝑋 × V) → 𝐴𝑋)
31, 2simplbiim 508 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ (𝑋 × V)) → 𝐴𝑋)
4 df-res 5537 . 2 (𝑅𝑋) = (𝑅 ∩ (𝑋 × V))
53, 4eleq2s 2851 1 (⟨𝐴, 𝐵⟩ ∈ (𝑅𝑋) → 𝐴𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114  Vcvv 3398  cin 3842  cop 4522   × cxp 5523  cres 5527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-opab 5093  df-xp 5531  df-res 5537
This theorem is referenced by:  bj-brresdm  34960
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