Theorem brelrn 5015

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 5013	. 2 |- ( ( A e. _V /\ B e. _V /\ A C B ) -> B e. ran C )
4	1, 2, 3	mp3an12 1368	1 |- ( A C B -> B e. ran C )

This proof depends on syntax axioms:   -> wi 4   e. wcel 2209   _Vcvv 2821   class class class wbr 4130   ran crn 4775

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-cnv 4782  df-dm 4784  df-rn 4785

This theorem is used by:  opelrn  5016  dfco2a  5288  cores  5291  dffun9  5406  funcnv  5442  rntpos  6528  tfrexlem  6605