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Theorem inxpssres 5701
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)) ⊆ (𝑅𝐴)

Proof of Theorem inxpssres
StepHypRef Expression
1 ssid 4005 . . . 4 𝐴𝐴
2 ssv 4007 . . . 4 𝐵 ⊆ V
3 xpss12 5699 . . . 4 ((𝐴𝐴𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V))
41, 2, 3mp2an 692 . . 3 (𝐴 × 𝐵) ⊆ (𝐴 × V)
5 sslin 4242 . . 3 ((𝐴 × 𝐵) ⊆ (𝐴 × V) → (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ∩ (𝐴 × V)))
64, 5ax-mp 5 . 2 (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅 ∩ (𝐴 × V))
7 df-res 5696 . 2 (𝑅𝐴) = (𝑅 ∩ (𝐴 × V))
86, 7sseqtrri 4032 1 (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3479  cin 3949  wss 3950   × cxp 5682  cres 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-in 3957  df-ss 3967  df-opab 5205  df-xp 5690  df-res 5696
This theorem is referenced by:  ssrnres  6197  idreseqidinxp  38311  refrelsredund4  38634  refrelredund4  38637
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