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Theorem inxpssres 5360
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 3849 . . . 4 𝐴𝐴
2 ssv 3851 . . . 4 𝐵 ⊆ V
3 xpss12 5358 . . . 4 ((𝐴𝐴𝐵 ⊆ V) → (𝐴 × 𝐵) ⊆ (𝐴 × V))
41, 2, 3mp2an 685 . . 3 (𝐴 × 𝐵) ⊆ (𝐴 × V)
5 sslin 4064 . . 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 5355 . 2 (𝑅𝐴) = (𝑅 ∩ (𝐴 × V))
86, 7sseqtr4i 3864 1 (𝑅 ∩ (𝐴 × 𝐵)) ⊆ (𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3415  cin 3798  wss 3799   × cxp 5341  cres 5345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-v 3417  df-in 3806  df-ss 3813  df-opab 4937  df-xp 5349  df-res 5355
This theorem is referenced by:  ssrnres  5814  idreseqidinxp  34630  refrelsred4  34922  refrelred4  34925
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