Theorem xrnrel 35177

Description: A range Cartesian product is a relation. This is Scott Fenton's txprel 32951 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 32951. (Contributed by Scott Fenton, 31-Mar-2012.)

Assertion

Ref	Expression
xrnrel	⊢ Rel (𝐴 ⋉ 𝐵)

Proof of Theorem xrnrel

Step	Hyp	Ref	Expression
1		xrnss3v 35176	. . 3 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V))
2		xpss 5466	. . 3 ⊢ (V × (V × V)) ⊆ (V × V)
3	1, 2	sstri 3904	. 2 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × V)
4		df-rel 5457	. 2 ⊢ (Rel (𝐴 ⋉ 𝐵) ↔ (𝐴 ⋉ 𝐵) ⊆ (V × V))
5	3, 4	mpbir 232	1 ⊢ Rel (𝐴 ⋉ 𝐵)

Syntax hints:  Vcvv 3440   ⊆ wss 3865   × cxp 5448  Rel wrel 5455   ⋉ cxrn 35005

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-br 4969  df-opab 5031  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-res 5462  df-xrn 35175

This theorem is referenced by:  dfxrn2  35180  elecxrn  35190  inxpxrn  35195  br1cnvxrn2  35196  disjxrn  35529