Theorem brelrn 4957

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 4955	. 2 |- ( ( A e. _V /\ B e. _V /\ A C B ) -> B e. ran C )
4	1, 2, 3	mp3an12 1361	1 |- ( A C B -> B e. ran C )

Syntax hints:   -> wi 4   e. wcel 2200   _Vcvv 2799   class class class wbr 4083   ran crn 4720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-cnv 4727  df-dm 4729  df-rn 4730

This theorem is referenced by:  opelrn  4958  dfco2a  5229  cores  5232  dffun9  5347  funcnv  5382  rntpos  6403  tfrexlem  6480