Theorem bj-opelresdm 35316

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 5897. (Contributed by BJ, 25-Dec-2023.)

Assertion

Ref	Expression
bj-opelresdm	⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋)

Proof of Theorem bj-opelresdm

Step	Hyp	Ref	Expression
1		elin 3903	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ (𝑋 × V)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × V)))
2		opelxp1 5630	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × V) → 𝐴 ∈ 𝑋)
3	1, 2	simplbiim 505	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ (𝑋 × V)) → 𝐴 ∈ 𝑋)
4		df-res 5601	. 2 ⊢ (𝑅 ↾ 𝑋) = (𝑅 ∩ (𝑋 × V))
5	3, 4	eleq2s 2857	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋)

Syntax hints:   → wi 4   ∈ wcel 2106  Vcvv 3432   ∩ cin 3886  ⟨cop 4567   × cxp 5587   ↾ cres 5591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-opab 5137  df-xp 5595  df-res 5601

This theorem is referenced by:  bj-brresdm  35317