Theorem brelrn 4821

Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 13-Aug-2004.)

Hypotheses

Ref	Expression
brelrn.1	|- A e. _V
brelrn.2	|- B e. _V

Assertion

Ref	Expression
brelrn	|- ( A C B -> B e. ran C )

Proof of Theorem brelrn

Step	Hyp	Ref	Expression
1		brelrn.1	. 2 |- A e. _V
2		brelrn.2	. 2 |- B e. _V
3		brelrng 4819	. 2 |- ( ( A e. _V /\ B e. _V /\ A C B ) -> B e. ran C )
4	1, 2, 3	mp3an12 1309	1 |- ( A C B -> B e. ran C )

Syntax hints:   -> wi 4   e. wcel 2128   _Vcvv 2712   class class class wbr 3967   ran crn 4589

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4028  df-cnv 4596  df-dm 4598  df-rn 4599

This theorem is referenced by:  opelrn  4822  dfco2a  5088  cores  5091  dffun9  5201  funcnv  5233  rntpos  6206  tfrexlem  6283