Theorem bj-opelresdm 38271

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 5988. (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 4096	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ (𝑋 × V)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑅 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × V)))
2		opelxp1 5698	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × V) → 𝐴 ∈ 𝑋)
3	1, 2	simplbiim 514	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ∩ (𝑋 × V)) → 𝐴 ∈ 𝑋)
4		df-res 5668	. 2 ⊢ (𝑅 ↾ 𝑋) = (𝑅 ∩ (𝑋 × V))
5	3, 4	eleq2s 2882	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑅 ↾ 𝑋) → 𝐴 ∈ 𝑋)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  Vcvv 3454   ∩ cin 4092  ⟨cop 4838   × cxp 5654   ↾ cres 5658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-opab 5172  df-xp 5662  df-res 5668

This theorem is used by:  bj-brresdm  38272