Mathbox for Peter Mazsa < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xrnrel Structured version   Visualization version   GIF version

Theorem xrnrel 36091
 Description: A range Cartesian product is a relation. This is Scott Fenton's txprel 33756 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 33756. (Contributed by Scott Fenton, 31-Mar-2012.)
Assertion
Ref Expression
xrnrel Rel (𝐴𝐵)

Proof of Theorem xrnrel
StepHypRef Expression
1 xrnss3v 36090 . . 3 (𝐴𝐵) ⊆ (V × (V × V))
2 xpss 5543 . . 3 (V × (V × V)) ⊆ (V × V)
31, 2sstri 3903 . 2 (𝐴𝐵) ⊆ (V × V)
4 df-rel 5534 . 2 (Rel (𝐴𝐵) ↔ (𝐴𝐵) ⊆ (V × V))
53, 4mpbir 234 1 Rel (𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:  Vcvv 3409   ⊆ wss 3860   × cxp 5525  Rel wrel 5532   ⋉ cxrn 35918 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 2113  ax-9 2121  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5036  df-opab 5098  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-res 5539  df-xrn 36089 This theorem is referenced by:  dfxrn2  36094  elecxrn  36104  inxpxrn  36109  br1cnvxrn2  36110  disjxrn  36443
 Copyright terms: Public domain W3C validator