Theorem xrnrel 37046

Description: A range Cartesian product is a relation. This is Scott Fenton's txprel 34679 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 34679. (Contributed by Scott Fenton, 31-Mar-2012.)

Assertion

Ref	Expression
xrnrel	⊢ Rel (𝐴 ⋉ 𝐵)

Proof of Theorem xrnrel

Step	Hyp	Ref	Expression
1		xrnss3v 37045	. . 3 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V))
2		xpss 5685	. . 3 ⊢ (V × (V × V)) ⊆ (V × V)
3	1, 2	sstri 3987	. 2 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × V)
4		df-rel 5676	. 2 ⊢ (Rel (𝐴 ⋉ 𝐵) ↔ (𝐴 ⋉ 𝐵) ⊆ (V × V))
5	3, 4	mpbir 230	1 ⊢ Rel (𝐴 ⋉ 𝐵)

Syntax hints:  Vcvv 3473   ⊆ wss 3944   × cxp 5667  Rel wrel 5674   ⋉ cxrn 36845

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-res 5681  df-xrn 37044

This theorem is referenced by:  dfxrn2  37049  elecxrn  37059  inxpxrn  37068  br1cnvxrn2  37069  disjxrn  37419  disjxrnres5  37420