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Theorem inxpssres 5570
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 3992 . . . 4 𝐴𝐴
2 ssv 3994 . . . 4 𝐵 ⊆ V
3 xpss12 5568 . . . 4 ((𝐴𝐴𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V))
41, 2, 3mp2an 688 . . 3 (𝐴 × 𝐵) ⊆ (𝐴 × V)
5 sslin 4214 . . 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 5565 . 2 (𝑅𝐴) = (𝑅 ∩ (𝐴 × V))
86, 7sseqtrri 4007 1 (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3499  cin 3938  wss 3939   × cxp 5551  cres 5555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-rab 3151  df-v 3501  df-in 3946  df-ss 3955  df-opab 5125  df-xp 5559  df-res 5565
This theorem is referenced by:  ssrnres  6032  idreseqidinxp  35437  refrelsredund4  35736  refrelredund4  35739
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