Theorem bj-opelresdm 37319

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 5943. (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 3916	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ (𝑋 × V)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × V)))
2		opelxp1 5665	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × V) → 𝐴 ∈ 𝑋)
3	1, 2	simplbiim 504	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ (𝑋 × V)) → 𝐴 ∈ 𝑋)
4		df-res 5635	. 2 ⊢ (𝑅 ↾ 𝑋) = (𝑅 ∩ (𝑋 × V))
5	3, 4	eleq2s 2853	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3439   ∩ cin 3899  ⟨cop 4585   × cxp 5621   ↾ cres 5625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-opab 5160  df-xp 5629  df-res 5635

This theorem is referenced by:  bj-brresdm  37320