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Theorem restrreld 36978
 Description: The restriction of a transitive relation is a transitive relation. (Contributed by Richard Penner, 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 5050 . . 3 (𝑅𝐴) = (𝑅 ∩ (𝐴 × V))
42, 3syl6eq 2660 . 2 (𝜑𝑆 = (𝑅 ∩ (𝐴 × V)))
51, 4xpintrreld 36977 1 (𝜑 → (𝑆𝑆) ⊆ 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540   × cxp 5036   ↾ cres 5040   ∘ ccom 5042 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050 This theorem is referenced by: (None)
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