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Theorem brelrng 4924
Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 29-Jun-2008.)
Assertion
Ref Expression
brelrng  |-  ( ( A  e.  F  /\  B  e.  G  /\  A C B )  ->  B  e.  ran  C )

Proof of Theorem brelrng
StepHypRef Expression
1 brcnvg 4878 . . . . 5  |-  ( ( B  e.  G  /\  A  e.  F )  ->  ( B `' C A 
<->  A C B ) )
21ancoms 439 . . . 4  |-  ( ( A  e.  F  /\  B  e.  G )  ->  ( B `' C A 
<->  A C B ) )
32biimp3ar 1282 . . 3  |-  ( ( A  e.  F  /\  B  e.  G  /\  A C B )  ->  B `' C A )
4 breldmg 4900 . . . 4  |-  ( ( B  e.  G  /\  A  e.  F  /\  B `' C A )  ->  B  e.  dom  `' C
)
543com12 1155 . . 3  |-  ( ( A  e.  F  /\  B  e.  G  /\  B `' C A )  ->  B  e.  dom  `' C
)
63, 5syld3an3 1227 . 2  |-  ( ( A  e.  F  /\  B  e.  G  /\  A C B )  ->  B  e.  dom  `' C
)
7 df-rn 4716 . 2  |-  ran  C  =  dom  `' C
86, 7syl6eleqr 2387 1  |-  ( ( A  e.  F  /\  B  e.  G  /\  A C B )  ->  B  e.  ran  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    e. wcel 1696   class class class wbr 4039   `'ccnv 4704   dom cdm 4705   ran crn 4706
This theorem is referenced by:  brelrn  4925  relelrn  4928  sossfld  5136  fvrn0  5566  pgpfaclem1  15332
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-cnv 4713  df-dm 4715  df-rn 4716
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