Theorem bj-opelresdm 36560

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 5985. (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 3960	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ (𝑋 × V)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × V)))
2		opelxp1 5714	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × V) → 𝐴 ∈ 𝑋)
3	1, 2	simplbiim 504	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ (𝑋 × V)) → 𝐴 ∈ 𝑋)
4		df-res 5684	. 2 ⊢ (𝑅 ↾ 𝑋) = (𝑅 ∩ (𝑋 × V))
5	3, 4	eleq2s 2846	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋)

Syntax hints:   → wi 4   ∈ wcel 2099  Vcvv 3469   ∩ cin 3943  ⟨cop 4630   × cxp 5670   ↾ cres 5674

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 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-opab 5205  df-xp 5678  df-res 5684

This theorem is referenced by:  bj-brresdm  36561