Theorem xrnrel 36430

Description: A range Cartesian product is a relation. This is Scott Fenton's txprel 34108 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 34108. (Contributed by Scott Fenton, 31-Mar-2012.)

Assertion

Ref	Expression
xrnrel	⊢ Rel (𝐴 ⋉ 𝐵)

Proof of Theorem xrnrel

Step	Hyp	Ref	Expression
1		xrnss3v 36429	. . 3 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V))
2		xpss 5596	. . 3 ⊢ (V × (V × V)) ⊆ (V × V)
3	1, 2	sstri 3926	. 2 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × V)
4		df-rel 5587	. 2 ⊢ (Rel (𝐴 ⋉ 𝐵) ↔ (𝐴 ⋉ 𝐵) ⊆ (V × V))
5	3, 4	mpbir 230	1 ⊢ Rel (𝐴 ⋉ 𝐵)

Syntax hints:  Vcvv 3422   ⊆ wss 3883   × cxp 5578  Rel wrel 5585   ⋉ cxrn 36259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-res 5592  df-xrn 36428

This theorem is referenced by:  dfxrn2  36433  elecxrn  36443  inxpxrn  36448  br1cnvxrn2  36449  disjxrn  36782