Theorem inxpssres 5693

Description: Intersection with a Cartesian product is a subclass of restriction. (Contributed by Peter Mazsa, 19-Jul-2019.)

Assertion

Ref	Expression
inxpssres	⊢ (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ↾ 𝐴)

Proof of Theorem inxpssres

Step	Hyp	Ref	Expression
1		ssid 4004	. . . 4 ⊢ 𝐴 ⊆ 𝐴
2		ssv 4006	. . . 4 ⊢ 𝐵 ⊆ V
3		xpss12 5691	. . . 4 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V))
4	1, 2, 3	mp2an 689	. . 3 ⊢ (𝐴 × 𝐵) ⊆ (𝐴 × V)
5		sslin 4234	. . 3 ⊢ ((𝐴 × 𝐵) ⊆ (𝐴 × V) → (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ∩ (𝐴 × V)))
6	4, 5	ax-mp 5	. 2 ⊢ (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ∩ (𝐴 × V))
7		df-res 5688	. 2 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × V))
8	6, 7	sseqtrri 4019	1 ⊢ (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ↾ 𝐴)

Syntax hints:  Vcvv 3473   ∩ cin 3947   ⊆ wss 3948   × cxp 5674   ↾ cres 5678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-in 3955  df-ss 3965  df-opab 5211  df-xp 5682  df-res 5688

This theorem is referenced by:  ssrnres  6177  idreseqidinxp  37482  refrelsredund4  37806  refrelredund4  37809