Theorem inxpssres 5664

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 3952	. . . 4 ⊢ 𝐴 ⊆ 𝐴
2		ssv 3954	. . . 4 ⊢ 𝐵 ⊆ V
3		xpss12 5662	. . . 4 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V))
4	1, 2, 3	mp2an 705	. . 3 ⊢ (𝐴 × 𝐵) ⊆ (𝐴 × V)
5		sslin 4187	. . 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 5659	. 2 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × V))
8	6, 7	sseqtrri 3979	1 ⊢ (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ↾ 𝐴)

This proof depends on syntax axioms:  Vcvv 3450   ∩ cin 3897   ⊆ wss 3898   × cxp 5645   ↾ cres 5649

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-in 3905  df-ss 3915  df-opab 5167  df-xp 5653  df-res 5659

This theorem is used by:  ssrnres  6165  idreseqidinxp  39167  refrelsredund4  39568  refrelredund4  39571