Theorem xrnrel 38845

Description: A range Cartesian product is a relation. This is Scott Fenton's txprel 36191 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 36191. (Contributed by Scott Fenton, 31-Mar-2012.)

Assertion

Ref	Expression
xrnrel	⊢ Rel (𝐴 ⋉ 𝐵)

Proof of Theorem xrnrel

Step	Hyp	Ref	Expression
1		xrnss3v 38844	. . 3 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V))
2		xpss 5661	. . 3 ⊢ (V × (V × V)) ⊆ (V × V)
3	1, 2	sstri 3945	. 2 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × V)
4		df-rel 5652	. 2 ⊢ (Rel (𝐴 ⋉ 𝐵) ↔ (𝐴 ⋉ 𝐵) ⊆ (V × V))
5	3, 4	mpbir 233	1 ⊢ Rel (𝐴 ⋉ 𝐵)

Syntax hints:  Vcvv 3453   ⊆ wss 3904   × cxp 5643  Rel wrel 5650   ⋉ cxrn 38637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-res 5657  df-xrn 38843

This theorem is referenced by:  dfxrn2  38848  elecxrn  38868  inxpxrn  38881  br1cnvxrn2  38882  disjxrn  39309  disjxrnres5  39310