Theorem xrnrel 35619

Description: A range Cartesian product is a relation. This is Scott Fenton's txprel 33335 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 33335. (Contributed by Scott Fenton, 31-Mar-2012.)

Assertion

Ref	Expression
xrnrel	⊢ Rel (𝐴 ⋉ 𝐵)

Proof of Theorem xrnrel

Step	Hyp	Ref	Expression
1		xrnss3v 35618	. . 3 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V))
2		xpss 5565	. . 3 ⊢ (V × (V × V)) ⊆ (V × V)
3	1, 2	sstri 3975	. 2 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × V)
4		df-rel 5556	. 2 ⊢ (Rel (𝐴 ⋉ 𝐵) ↔ (𝐴 ⋉ 𝐵) ⊆ (V × V))
5	3, 4	mpbir 233	1 ⊢ Rel (𝐴 ⋉ 𝐵)

Syntax hints:  Vcvv 3494   ⊆ wss 3935   × cxp 5547  Rel wrel 5554   ⋉ cxrn 35446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-res 5561  df-xrn 35617

This theorem is referenced by:  dfxrn2  35622  elecxrn  35632  inxpxrn  35637  br1cnvxrn2  35638  disjxrn  35971