Theorem xrnrel 38362

Description: A range Cartesian product is a relation. This is Scott Fenton's txprel 35874 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 35874. (Contributed by Scott Fenton, 31-Mar-2012.)

Assertion

Ref	Expression
xrnrel	⊢ Rel (𝐴 ⋉ 𝐵)

Proof of Theorem xrnrel

Step	Hyp	Ref	Expression
1		xrnss3v 38361	. . 3 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V))
2		xpss 5657	. . 3 ⊢ (V × (V × V)) ⊆ (V × V)
3	1, 2	sstri 3959	. 2 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × V)
4		df-rel 5648	. 2 ⊢ (Rel (𝐴 ⋉ 𝐵) ↔ (𝐴 ⋉ 𝐵) ⊆ (V × V))
5	3, 4	mpbir 231	1 ⊢ Rel (𝐴 ⋉ 𝐵)

Syntax hints:  Vcvv 3450   ⊆ wss 3917   × cxp 5639  Rel wrel 5646   ⋉ cxrn 38175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-res 5653  df-xrn 38360

This theorem is referenced by:  dfxrn2  38365  elecxrn  38379  inxpxrn  38388  br1cnvxrn2  38389  disjxrn  38745  disjxrnres5  38746