Theorem inxpssres 5552

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 3909	. . . 4 ⊢ 𝐴 ⊆ 𝐴
2		ssv 3911	. . . 4 ⊢ 𝐵 ⊆ V
3		xpss12 5550	. . . 4 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V))
4	1, 2, 3	mp2an 692	. . 3 ⊢ (𝐴 × 𝐵) ⊆ (𝐴 × V)
5		sslin 4135	. . 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 5547	. 2 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × V))
8	6, 7	sseqtrri 3924	1 ⊢ (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ↾ 𝐴)

Syntax hints:  Vcvv 3400   ∩ cin 3852   ⊆ wss 3853   × cxp 5533   ↾ cres 5537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-rab 3063  df-v 3402  df-in 3860  df-ss 3870  df-opab 5103  df-xp 5541  df-res 5547

This theorem is referenced by:  ssrnres  6020  idreseqidinxp  36101  refrelsredund4  36401  refrelredund4  36404