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Theorem xpintrreld 39380
 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 14201 . . 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 39379 1 (𝜑 → (𝑆𝑆) ⊆ 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1507   ∩ cin 3828   ⊆ wss 3829   × cxp 5405   ∘ ccom 5411 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-br 4930  df-opab 4992  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419 This theorem is referenced by:  restrreld  39381
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