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 (𝐴 ⋉ 𝐵)

Proof of Theorem xrnrel

Step	Hyp	Ref	Expression
1		xrnss3v 36090	. . 3 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V))
2		xpss 5543	. . 3 ⊢ (V × (V × V)) ⊆ (V × V)
3	1, 2	sstri 3903	. 2 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × V)
4		df-rel 5534	. 2 ⊢ (Rel (𝐴 ⋉ 𝐵) ↔ (𝐴 ⋉ 𝐵) ⊆ (V × V))
5	3, 4	mpbir 234	1 ⊢ Rel (𝐴 ⋉ 𝐵)

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