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Theorem xpintrreld 43569
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 15025 . . 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 43568 1 (𝜑 → (𝑆𝑆) ⊆ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  cin 3969  wss 3970   × cxp 5697  ccom 5703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5170  df-opab 5232  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711
This theorem is referenced by:  restrreld  43570
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