Theorem xrnrel 39573

Description: A range Cartesian product is a relation. This is Scott Fenton's txprel 36849 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 36849. (Contributed by Scott Fenton, 31-Mar-2012.)

Assertion

Ref	Expression
xrnrel	⊢ Rel (𝐴 ⋉ 𝐵)

Proof of Theorem xrnrel

Step	Hyp	Ref	Expression
1		xrnss3v 39572	. . 3 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V))
2		xpss 5672	. . 3 ⊢ (V × (V × V)) ⊆ (V × V)
3	1, 2	sstri 3922	. 2 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × V)
4		df-rel 5663	. 2 ⊢ (Rel (𝐴 ⋉ 𝐵) ↔ (𝐴 ⋉ 𝐵) ⊆ (V × V))
5	3, 4	mpbir 234	1 ⊢ Rel (𝐴 ⋉ 𝐵)

This proof depends on syntax axioms:  Vcvv 3454   ⊆ wss 3900   × cxp 5654  Rel wrel 5661   ⋉ cxrn 39365

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-res 5668  df-xrn 39571

This theorem is used by:  dfxrn2  39576  elecxrn  39596  inxpxrn  39609  br1cnvxrn2  39610  disjxrn  40037  disjxrnres5  40038