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Theorem xrnrel 38374
Description: A range Cartesian product is a relation. This is Scott Fenton's txprel 35880 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 35880. (Contributed by Scott Fenton, 31-Mar-2012.)
Assertion
Ref Expression
xrnrel Rel (𝐴𝐵)

Proof of Theorem xrnrel
StepHypRef Expression
1 xrnss3v 38373 . . 3 (𝐴𝐵) ⊆ (V × (V × V))
2 xpss 5701 . . 3 (V × (V × V)) ⊆ (V × V)
31, 2sstri 3993 . 2 (𝐴𝐵) ⊆ (V × V)
4 df-rel 5692 . 2 (Rel (𝐴𝐵) ↔ (𝐴𝐵) ⊆ (V × V))
53, 4mpbir 231 1 Rel (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3480  wss 3951   × cxp 5683  Rel wrel 5690  cxrn 38181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-res 5697  df-xrn 38372
This theorem is referenced by:  dfxrn2  38377  elecxrn  38387  inxpxrn  38396  br1cnvxrn2  38397  disjxrn  38747  disjxrnres5  38748
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