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Theorem xpintrreld 37478
Description: The intersection of a transitive relation with a cross product is a transitve relation. (Contributed by Richard Penner, 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 13669 . . 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 37477 1 (𝜑 → (𝑆𝑆) ⊆ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  cin 3559  wss 3560   × cxp 5082  ccom 5088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096
This theorem is referenced by:  restrreld  37479
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