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Theorem inxpssres 5706
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 4018 . . . 4 𝐴𝐴
2 ssv 4020 . . . 4 𝐵 ⊆ V
3 xpss12 5704 . . . 4 ((𝐴𝐴𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V))
41, 2, 3mp2an 692 . . 3 (𝐴 × 𝐵) ⊆ (𝐴 × V)
5 sslin 4251 . . 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 5701 . 2 (𝑅𝐴) = (𝑅 ∩ (𝐴 × V))
86, 7sseqtrri 4033 1 (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3478  cin 3962  wss 3963   × cxp 5687  cres 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-in 3970  df-ss 3980  df-opab 5211  df-xp 5695  df-res 5701
This theorem is referenced by:  ssrnres  6200  idreseqidinxp  38291  refrelsredund4  38614  refrelredund4  38617
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