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Theorem brelrng 5180
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 5131 . . . . 5  |-  ( ( B  e.  G  /\  A  e.  F )  ->  ( B `' C A 
<->  A C B ) )
21ancoms 453 . . . 4  |-  ( ( A  e.  F  /\  B  e.  G )  ->  ( B `' C A 
<->  A C B ) )
32biimp3ar 1320 . . 3  |-  ( ( A  e.  F  /\  B  e.  G  /\  A C B )  ->  B `' C A )
4 breldmg 5156 . . . 4  |-  ( ( B  e.  G  /\  A  e.  F  /\  B `' C A )  ->  B  e.  dom  `' C
)
543com12 1192 . . 3  |-  ( ( A  e.  F  /\  B  e.  G  /\  B `' C A )  ->  B  e.  dom  `' C
)
63, 5syld3an3 1264 . 2  |-  ( ( A  e.  F  /\  B  e.  G  /\  A C B )  ->  B  e.  dom  `' C
)
7 df-rn 4962 . 2  |-  ran  C  =  dom  `' C
86, 7syl6eleqr 2553 1  |-  ( ( A  e.  F  /\  B  e.  G  /\  A C B )  ->  B  e.  ran  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965    e. wcel 1758   class class class wbr 4403   `'ccnv 4950   dom cdm 4951   ran crn 4952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-cnv 4959  df-dm 4961  df-rn 4962
This theorem is referenced by:  brelrn  5181  relelrn  5184  sossfld  5396  fvrn0  5824  pgpfaclem1  16714  perpln2  23255
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