Metamath Proof Explorer


Theorem xrnrel

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

Ref Expression
Assertion xrnrel
|- Rel ( A |X. B )

Proof

Step Hyp Ref Expression
1 xrnss3v
 |-  ( A |X. B ) C_ ( _V X. ( _V X. _V ) )
2 xpss
 |-  ( _V X. ( _V X. _V ) ) C_ ( _V X. _V )
3 1 2 sstri
 |-  ( A |X. B ) C_ ( _V X. _V )
4 df-rel
 |-  ( Rel ( A |X. B ) <-> ( A |X. B ) C_ ( _V X. _V ) )
5 3 4 mpbir
 |-  Rel ( A |X. B )