Theorem xrnrel 34136

Description: A range Cartesian product is a relation. This is Scott Fenton's txprel 31986 with different symbols, cf. https://github.com/metamath/set.mm/issues/2469 . (Contributed by Scott Fenton, 31-Mar-2012.)

Assertion

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

Proof of Theorem xrnrel

Step	Hyp	Ref	Expression
1		xrnss3v 34135	. . 3 |- ( A |X. B ) C_ ( _V X. ( _V X. _V ) )
2		xpss 5226	. . 3 |- ( _V X. ( _V X. _V ) ) C_ ( _V X. _V )
3	1, 2	sstri 3612	. 2 |- ( A |X. B ) C_ ( _V X. _V )
4		df-rel 5121	. 2 |- ( Rel ( A |X. B ) <-> ( A |X. B ) C_ ( _V X. _V ) )
5	3, 4	mpbir 221	1 |- Rel ( A |X. B )

Syntax hints:   _Vcvv 3200   C_ wss 3574   X. cxp 5112   Rel wrel 5119   |X. cxrn 33982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-res 5126  df-xrn 34134

This theorem is referenced by: (None)