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Theorem dmfcoafv 38091
Description: Domains of a function composition, analogous to dmfco 5955. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
Assertion
Ref Expression
dmfcoafv  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( A  e.  dom  ( F  o.  G
)  <->  ( G''' A )  e.  dom  F ) )

Proof of Theorem dmfcoafv
StepHypRef Expression
1 dmfco 5955 . 2  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( A  e.  dom  ( F  o.  G
)  <->  ( G `  A )  e.  dom  F ) )
2 funres 5640 . . . . . . 7  |-  ( Fun 
G  ->  Fun  ( G  |`  { A } ) )
32anim2i 571 . . . . . 6  |-  ( ( A  e.  dom  G  /\  Fun  G )  -> 
( A  e.  dom  G  /\  Fun  ( G  |`  { A } ) ) )
43ancoms 454 . . . . 5  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( A  e.  dom  G  /\  Fun  ( G  |`  { A } ) ) )
5 df-dfat 38032 . . . . . 6  |-  ( G defAt 
A  <->  ( A  e. 
dom  G  /\  Fun  ( G  |`  { A }
) ) )
6 afvfundmfveq 38054 . . . . . 6  |-  ( G defAt 
A  ->  ( G''' A )  =  ( G `
 A ) )
75, 6sylbir 216 . . . . 5  |-  ( ( A  e.  dom  G  /\  Fun  ( G  |`  { A } ) )  ->  ( G''' A )  =  ( G `  A ) )
84, 7syl 17 . . . 4  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( G''' A )  =  ( G `  A ) )
98eqcomd 2437 . . 3  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( G `  A
)  =  ( G''' A ) )
109eleq1d 2498 . 2  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( ( G `  A )  e.  dom  F  <-> 
( G''' A )  e.  dom  F ) )
111, 10bitrd 256 1  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( A  e.  dom  ( F  o.  G
)  <->  ( G''' A )  e.  dom  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   {csn 4002   dom cdm 4854    |` cres 4856    o. ccom 4858   Fun wfun 5595   ` cfv 5601   defAt wdfat 38029  '''cafv 38030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-res 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-dfat 38032  df-afv 38033
This theorem is referenced by: (None)
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