Theorem inxpssres 5572

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 3989	. . . 4 ⊢ 𝐴 ⊆ 𝐴
2		ssv 3991	. . . 4 ⊢ 𝐵 ⊆ V
3		xpss12 5570	. . . 4 ⊢ ((𝐴 ⊆ 𝐴 ∧ 𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V))
4	1, 2, 3	mp2an 690	. . 3 ⊢ (𝐴 × 𝐵) ⊆ (𝐴 × V)
5		sslin 4211	. . 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 5567	. 2 ⊢ (𝑅 ↾ 𝐴) = (𝑅 ∩ (𝐴 × V))
8	6, 7	sseqtrri 4004	1 ⊢ (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ↾ 𝐴)

Syntax hints:  Vcvv 3494   ∩ cin 3935   ⊆ wss 3936   × cxp 5553   ↾ cres 5557

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-in 3943  df-ss 3952  df-opab 5129  df-xp 5561  df-res 5567

This theorem is referenced by:  ssrnres  6035  idreseqidinxp  35582  refrelsredund4  35882  refrelredund4  35885