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Theorem bj-opelresdm 34429
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 5852. (Contributed by BJ, 25-Dec-2023.)
Assertion
Ref Expression
bj-opelresdm (⟨𝐴, 𝐵⟩ ∈ (𝑅𝑋) → 𝐴𝑋)

Proof of Theorem bj-opelresdm
StepHypRef Expression
1 elin 4167 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ (𝑋 × V)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × V)))
2 opelxp1 5589 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝑋 × V) → 𝐴𝑋)
31, 2simplbiim 507 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ (𝑋 × V)) → 𝐴𝑋)
4 df-res 5560 . 2 (𝑅𝑋) = (𝑅 ∩ (𝑋 × V))
53, 4eleq2s 2929 1 (⟨𝐴, 𝐵⟩ ∈ (𝑅𝑋) → 𝐴𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  Vcvv 3493  cin 3933  cop 4565   × cxp 5546  cres 5550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-opab 5120  df-xp 5554  df-res 5560
This theorem is referenced by:  bj-brresdm  34430
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