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Theorem xpintrreld 41274
Description: The intersection of a transitive relation with a Cartesian product is a transitive relation. (Contributed by RP, 24-Dec-2019.)
Hypotheses
Ref Expression
xpintrreld.r (𝜑 → (𝑅𝑅) ⊆ 𝑅)
xpintrreld.s (𝜑𝑆 = (𝑅 ∩ (𝐴 × 𝐵)))
Assertion
Ref Expression
xpintrreld (𝜑 → (𝑆𝑆) ⊆ 𝑆)

Proof of Theorem xpintrreld
StepHypRef Expression
1 xpintrreld.r . 2 (𝜑 → (𝑅𝑅) ⊆ 𝑅)
2 xptrrel 14691 . . 3 ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
32a1i 11 . 2 (𝜑 → ((𝐴 × 𝐵) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵))
4 xpintrreld.s . 2 (𝜑𝑆 = (𝑅 ∩ (𝐴 × 𝐵)))
51, 3, 4trrelind 41273 1 (𝜑 → (𝑆𝑆) ⊆ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  cin 3886  wss 3887   × cxp 5587  ccom 5593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601
This theorem is referenced by:  restrreld  41275
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