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Theorem restrreld 42408
Description: The restriction of a transitive relation is a transitive relation. (Contributed by RP, 24-Dec-2019.)
Hypotheses
Ref Expression
restrreld.r (𝜑 → (𝑅𝑅) ⊆ 𝑅)
restrreld.s (𝜑𝑆 = (𝑅𝐴))
Assertion
Ref Expression
restrreld (𝜑 → (𝑆𝑆) ⊆ 𝑆)

Proof of Theorem restrreld
StepHypRef Expression
1 restrreld.r . 2 (𝜑 → (𝑅𝑅) ⊆ 𝑅)
2 restrreld.s . . 3 (𝜑𝑆 = (𝑅𝐴))
3 df-res 5688 . . 3 (𝑅𝐴) = (𝑅 ∩ (𝐴 × V))
42, 3eqtrdi 2788 . 2 (𝜑𝑆 = (𝑅 ∩ (𝐴 × V)))
51, 4xpintrreld 42407 1 (𝜑 → (𝑆𝑆) ⊆ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  Vcvv 3474  cin 3947  wss 3948   × cxp 5674  cres 5678  ccom 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688
This theorem is referenced by: (None)
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