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Theorem brelrng 4817
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 4767 . . . . 5  |-  ( ( B  e.  G  /\  A  e.  F )  ->  ( B `' C A 
<->  A C B ) )
21ancoms 266 . . . 4  |-  ( ( A  e.  F  /\  B  e.  G )  ->  ( B `' C A 
<->  A C B ) )
32biimp3ar 1328 . . 3  |-  ( ( A  e.  F  /\  B  e.  G  /\  A C B )  ->  B `' C A )
4 breldmg 4792 . . . 4  |-  ( ( B  e.  G  /\  A  e.  F  /\  B `' C A )  ->  B  e.  dom  `' C
)
543com12 1189 . . 3  |-  ( ( A  e.  F  /\  B  e.  G  /\  B `' C A )  ->  B  e.  dom  `' C
)
63, 5syld3an3 1265 . 2  |-  ( ( A  e.  F  /\  B  e.  G  /\  A C B )  ->  B  e.  dom  `' C
)
7 df-rn 4597 . 2  |-  ran  C  =  dom  `' C
86, 7eleqtrrdi 2251 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 104    /\ w3a 963    e. wcel 2128   class class class wbr 3965   `'ccnv 4585   dom cdm 4586   ran crn 4587
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 4082  ax-pow 4135  ax-pr 4169
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 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-cnv 4594  df-dm 4596  df-rn 4597
This theorem is referenced by:  opelrng  4818  brelrn  4819  relelrn  4822  fvssunirng  5483  shftfvalg  10718  ovshftex  10719  shftfval  10721
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