Theorem xrnrel 38426

Description: A range Cartesian product is a relation. This is Scott Fenton's txprel 35942 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 35942. (Contributed by Scott Fenton, 31-Mar-2012.)

Assertion

Ref	Expression
xrnrel	⊢ Rel (𝐴 ⋉ 𝐵)

Proof of Theorem xrnrel

Step	Hyp	Ref	Expression
1		xrnss3v 38425	. . 3 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V))
2		xpss 5635	. . 3 ⊢ (V × (V × V)) ⊆ (V × V)
3	1, 2	sstri 3940	. 2 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × V)
4		df-rel 5626	. 2 ⊢ (Rel (𝐴 ⋉ 𝐵) ↔ (𝐴 ⋉ 𝐵) ⊆ (V × V))
5	3, 4	mpbir 231	1 ⊢ Rel (𝐴 ⋉ 𝐵)

Syntax hints:  Vcvv 3437   ⊆ wss 3898   × cxp 5617  Rel wrel 5624   ⋉ cxrn 38234

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 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

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 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-res 5631  df-xrn 38424

This theorem is referenced by:  dfxrn2  38429  elecxrn  38449  inxpxrn  38462  br1cnvxrn2  38463  disjxrn  38864  disjxrnres5  38865