Theorem xrnrel 36529

Description: A range Cartesian product is a relation. This is Scott Fenton's txprel 34209 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 34209. (Contributed by Scott Fenton, 31-Mar-2012.)

Assertion

Ref	Expression
xrnrel	⊢ Rel (𝐴 ⋉ 𝐵)

Proof of Theorem xrnrel

Step	Hyp	Ref	Expression
1		xrnss3v 36528	. . 3 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V))
2		xpss 5607	. . 3 ⊢ (V × (V × V)) ⊆ (V × V)
3	1, 2	sstri 3932	. 2 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × V)
4		df-rel 5598	. 2 ⊢ (Rel (𝐴 ⋉ 𝐵) ↔ (𝐴 ⋉ 𝐵) ⊆ (V × V))
5	3, 4	mpbir 230	1 ⊢ Rel (𝐴 ⋉ 𝐵)

Syntax hints:  Vcvv 3434   ⊆ wss 3889   × cxp 5589  Rel wrel 5596   ⋉ cxrn 36360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-sn 4565  df-pr 4567  df-op 4571  df-br 5078  df-opab 5140  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-res 5603  df-xrn 36527

This theorem is referenced by:  dfxrn2  36532  elecxrn  36542  inxpxrn  36547  br1cnvxrn2  36548  disjxrn  36881