Theorem relelrn 4770

Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 2-Jul-2008.)

Assertion

Ref	Expression
relelrn	|- ( ( Rel R /\ A R B ) -> B e. ran R )

Proof of Theorem relelrn

Step	Hyp	Ref	Expression
1		brrelex 4574	. 2 |- ( ( Rel R /\ A R B ) -> A e. _V )
2		brrelex2 4575	. 2 |- ( ( Rel R /\ A R B ) -> B e. _V )
3		simpr 109	. 2 |- ( ( Rel R /\ A R B ) -> A R B )
4		brelrng 4765	. 2 |- ( ( A e. _V /\ B e. _V /\ A R B ) -> B e. ran R )
5	1, 2, 3, 4	syl3anc 1216	1 |- ( ( Rel R /\ A R B ) -> B e. ran R )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1480   _Vcvv 2681   class class class wbr 3924   ran crn 4535   Rel wrel 4539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542  df-dm 4544  df-rn 4545

This theorem is referenced by:  relelrnb  4772  relelrni  4774  relfvssunirn  5430