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Theorem restrreld 43162
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 5684 . . 3 (𝑅𝐴) = (𝑅 ∩ (𝐴 × V))
42, 3eqtrdi 2781 . 2 (𝜑𝑆 = (𝑅 ∩ (𝐴 × V)))
51, 4xpintrreld 43161 1 (𝜑 → (𝑆𝑆) ⊆ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  Vcvv 3463  cin 3938  wss 3939   × cxp 5670  cres 5674  ccom 5676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684
This theorem is referenced by: (None)
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