Theorem xrnrel 37707

Description: A range Cartesian product is a relation. This is Scott Fenton's txprel 35321 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 35321. (Contributed by Scott Fenton, 31-Mar-2012.)

Assertion

Ref	Expression
xrnrel	⊢ Rel (𝐴 ⋉ 𝐵)

Proof of Theorem xrnrel

Step	Hyp	Ref	Expression
1		xrnss3v 37706	. . 3 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V))
2		xpss 5692	. . 3 ⊢ (V × (V × V)) ⊆ (V × V)
3	1, 2	sstri 3991	. 2 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × V)
4		df-rel 5683	. 2 ⊢ (Rel (𝐴 ⋉ 𝐵) ↔ (𝐴 ⋉ 𝐵) ⊆ (V × V))
5	3, 4	mpbir 230	1 ⊢ Rel (𝐴 ⋉ 𝐵)

Syntax hints:  Vcvv 3473   ⊆ wss 3948   × cxp 5674  Rel wrel 5681   ⋉ cxrn 37506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-res 5688  df-xrn 37705

This theorem is referenced by:  dfxrn2  37710  elecxrn  37720  inxpxrn  37729  br1cnvxrn2  37730  disjxrn  38080  disjxrnres5  38081