Theorem xrnrel 38400

Description: A range Cartesian product is a relation. This is Scott Fenton's txprel 35912 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 35912. (Contributed by Scott Fenton, 31-Mar-2012.)

Assertion

Ref	Expression
xrnrel	⊢ Rel (𝐴 ⋉ 𝐵)

Proof of Theorem xrnrel

Step	Hyp	Ref	Expression
1		xrnss3v 38399	. . 3 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V))
2		xpss 5632	. . 3 ⊢ (V × (V × V)) ⊆ (V × V)
3	1, 2	sstri 3944	. 2 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × V)
4		df-rel 5623	. 2 ⊢ (Rel (𝐴 ⋉ 𝐵) ↔ (𝐴 ⋉ 𝐵) ⊆ (V × V))
5	3, 4	mpbir 231	1 ⊢ Rel (𝐴 ⋉ 𝐵)

Syntax hints:  Vcvv 3436   ⊆ wss 3902   × cxp 5614  Rel wrel 5621   ⋉ cxrn 38213

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-res 5628  df-xrn 38398

This theorem is referenced by:  dfxrn2  38403  elecxrn  38417  inxpxrn  38426  br1cnvxrn2  38427  disjxrn  38783  disjxrnres5  38784