Theorem relelrn 4783

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 4587	. 2 |- ( ( Rel R /\ A R B ) -> A e. _V )
2		brrelex2 4588	. 2 |- ( ( Rel R /\ A R B ) -> B e. _V )
3		simpr 109	. 2 |- ( ( Rel R /\ A R B ) -> A R B )
4		brelrng 4778	. 2 |- ( ( A e. _V /\ B e. _V /\ A R B ) -> B e. ran R )
5	1, 2, 3, 4	syl3anc 1217	1 |- ( ( Rel R /\ A R B ) -> B e. ran R )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1481   _Vcvv 2689   class class class wbr 3937   ran crn 4548   Rel wrel 4552

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-dm 4557  df-rn 4558

This theorem is referenced by:  relelrnb  4785  relelrni  4787  relfvssunirn  5445