Theorem xrnrel 38756

Description: A range Cartesian product is a relation. This is Scott Fenton's txprel 36112 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 36112. (Contributed by Scott Fenton, 31-Mar-2012.)

Assertion

Ref	Expression
xrnrel	⊢ Rel (𝐴 ⋉ 𝐵)

Proof of Theorem xrnrel

Step	Hyp	Ref	Expression
1		xrnss3v 38755	. . 3 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V))
2		xpss 5641	. . 3 ⊢ (V × (V × V)) ⊆ (V × V)
3	1, 2	sstri 3931	. 2 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × V)
4		df-rel 5632	. 2 ⊢ (Rel (𝐴 ⋉ 𝐵) ↔ (𝐴 ⋉ 𝐵) ⊆ (V × V))
5	3, 4	mpbir 232	1 ⊢ Rel (𝐴 ⋉ 𝐵)

Syntax hints:  Vcvv 3432   ⊆ wss 3890   × cxp 5623  Rel wrel 5630   ⋉ cxrn 38548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-res 5637  df-xrn 38754

This theorem is referenced by:  dfxrn2  38759  elecxrn  38779  inxpxrn  38792  br1cnvxrn2  38793  disjxrn  39220  disjxrnres5  39221