Theorem xrnrel 39130

Description: A range Cartesian product is a relation. This is Scott Fenton's txprel 36456 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 36456. (Contributed by Scott Fenton, 31-Mar-2012.)

Assertion

Ref	Expression
xrnrel	⊢ Rel (𝐴 ⋉ 𝐵)

Proof of Theorem xrnrel

Step	Hyp	Ref	Expression
1		xrnss3v 39129	. . 3 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V))
2		xpss 5671	. . 3 ⊢ (V × (V × V)) ⊆ (V × V)
3	1, 2	sstri 3940	. 2 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × V)
4		df-rel 5662	. 2 ⊢ (Rel (𝐴 ⋉ 𝐵) ↔ (𝐴 ⋉ 𝐵) ⊆ (V × V))
5	3, 4	mpbir 234	1 ⊢ Rel (𝐴 ⋉ 𝐵)

This proof depends on syntax axioms:  Vcvv 3450   ⊆ wss 3899   × cxp 5653  Rel wrel 5660   ⋉ cxrn 38922

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-res 5667  df-xrn 39128

This theorem is used by:  dfxrn2  39133  elecxrn  39153  inxpxrn  39166  br1cnvxrn2  39167  disjxrn  39594  disjxrnres5  39595