Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  restrreld Structured version   Visualization version   GIF version

Theorem restrreld 40355
 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 5535 . . 3 (𝑅𝐴) = (𝑅 ∩ (𝐴 × V))
42, 3eqtrdi 2852 . 2 (𝜑𝑆 = (𝑅 ∩ (𝐴 × V)))
51, 4xpintrreld 40354 1 (𝜑 → (𝑆𝑆) ⊆ 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  Vcvv 3444   ∩ cin 3883   ⊆ wss 3884   × cxp 5521   ↾ cres 5525   ∘ ccom 5527 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator