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Theorem brelrng 4738
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 4688 . . . . 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 1307 . . 3  |-  ( ( A  e.  F  /\  B  e.  G  /\  A C B )  ->  B `' C A )
4 breldmg 4713 . . . 4  |-  ( ( B  e.  G  /\  A  e.  F  /\  B `' C A )  ->  B  e.  dom  `' C
)
543com12 1168 . . 3  |-  ( ( A  e.  F  /\  B  e.  G  /\  B `' C A )  ->  B  e.  dom  `' C
)
63, 5syld3an3 1244 . 2  |-  ( ( A  e.  F  /\  B  e.  G  /\  A C B )  ->  B  e.  dom  `' C
)
7 df-rn 4518 . 2  |-  ran  C  =  dom  `' C
86, 7syl6eleqr 2209 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 945    e. wcel 1463   class class class wbr 3897   `'ccnv 4506   dom cdm 4507   ran crn 4508
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-cnv 4515  df-dm 4517  df-rn 4518
This theorem is referenced by:  opelrng  4739  brelrn  4740  relelrn  4743  fvssunirng  5402  shftfvalg  10541  ovshftex  10542  shftfval  10544
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