Theorem brelrn 4899

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 4897	. 2 |- ( ( A e. _V /\ B e. _V /\ A C B ) -> B e. ran C )
4	1, 2, 3	mp3an12 1338	1 |- ( A C B -> B e. ran C )

Syntax hints:   -> wi 4   e. wcel 2167   _Vcvv 2763   class class class wbr 4033   ran crn 4664

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-cnv 4671  df-dm 4673  df-rn 4674

This theorem is referenced by:  opelrn  4900  dfco2a  5170  cores  5173  dffun9  5287  funcnv  5319  rntpos  6315  tfrexlem  6392