Theorem inxpssres 5678

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 3958	. . . 4 ⊢ 𝐴 ⊆ 𝐴
2		ssv 3960	. . . 4 ⊢ 𝐵 ⊆ V
3		xpss12 5676	. . . 4 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V))
4	1, 2, 3	mp2an 704	. . 3 ⊢ (𝐴 × 𝐵) ⊆ (𝐴 × V)
5		sslin 4194	. . 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 5673	. 2 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × V))
8	6, 7	sseqtrri 3985	1 ⊢ (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ↾ 𝐴)

Syntax hints:  Vcvv 3453   ∩ cin 3903   ⊆ wss 3904   × cxp 5659   ↾ cres 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3415  df-v 3455  df-in 3911  df-ss 3921  df-opab 5173  df-xp 5667  df-res 5673

This theorem is referenced by:  ssrnres  6176  idreseqidinxp  38910  refrelsredund4  39311  refrelredund4  39314