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Theorem brelrng 4593
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 4544 . . . . 5  |-  ( ( B  e.  G  /\  A  e.  F )  ->  ( B `' C A 
<->  A C B ) )
21ancoms 264 . . . 4  |-  ( ( A  e.  F  /\  B  e.  G )  ->  ( B `' C A 
<->  A C B ) )
32biimp3ar 1278 . . 3  |-  ( ( A  e.  F  /\  B  e.  G  /\  A C B )  ->  B `' C A )
4 breldmg 4569 . . . 4  |-  ( ( B  e.  G  /\  A  e.  F  /\  B `' C A )  ->  B  e.  dom  `' C
)
543com12 1143 . . 3  |-  ( ( A  e.  F  /\  B  e.  G  /\  B `' C A )  ->  B  e.  dom  `' C
)
63, 5syld3an3 1215 . 2  |-  ( ( A  e.  F  /\  B  e.  G  /\  A C B )  ->  B  e.  dom  `' C
)
7 df-rn 4382 . 2  |-  ran  C  =  dom  `' C
86, 7syl6eleqr 2173 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 103    /\ w3a 920    e. wcel 1434   class class class wbr 3793   `'ccnv 4370   dom cdm 4371   ran crn 4372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-cnv 4379  df-dm 4381  df-rn 4382
This theorem is referenced by:  opelrng  4594  brelrn  4595  relelrn  4598  fvssunirng  5221  shftfvalg  9844  ovshftex  9845  shftfval  9847
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