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Theorem inxpssres 5552
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 3909 . . . 4 𝐴𝐴
2 ssv 3911 . . . 4 𝐵 ⊆ V
3 xpss12 5550 . . . 4 ((𝐴𝐴𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V))
41, 2, 3mp2an 692 . . 3 (𝐴 × 𝐵) ⊆ (𝐴 × V)
5 sslin 4135 . . 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 5547 . 2 (𝑅𝐴) = (𝑅 ∩ (𝐴 × V))
86, 7sseqtrri 3924 1 (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3400  cin 3852  wss 3853   × cxp 5533  cres 5537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-rab 3063  df-v 3402  df-in 3860  df-ss 3870  df-opab 5103  df-xp 5541  df-res 5547
This theorem is referenced by:  ssrnres  6020  idreseqidinxp  36101  refrelsredund4  36401  refrelredund4  36404
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