Theorem bj-opelresdm 36677

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 5986. (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 3957	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ (𝑋 × V)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × V)))
2		opelxp1 5715	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × V) → 𝐴 ∈ 𝑋)
3	1, 2	simplbiim 503	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ (𝑋 × V)) → 𝐴 ∈ 𝑋)
4		df-res 5685	. 2 ⊢ (𝑅 ↾ 𝑋) = (𝑅 ∩ (𝑋 × V))
5	3, 4	eleq2s 2843	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋)

Syntax hints:   → wi 4   ∈ wcel 2098  Vcvv 3463   ∩ cin 3940  ⟨cop 4631   × cxp 5671   ↾ cres 5675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-opab 5207  df-xp 5679  df-res 5685

This theorem is referenced by:  bj-brresdm  36678