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Theorem dmfcoafv 28047
Description: Domains of a function composition, analogous to dmfco 5595. (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 5595 . 2  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( A  e.  dom  ( F  o.  G
)  <->  ( G `  A )  e.  dom  F ) )
2 funres 5295 . . . . . . 7  |-  ( Fun 
G  ->  Fun  ( G  |`  { A } ) )
32anim2i 552 . . . . . 6  |-  ( ( A  e.  dom  G  /\  Fun  G )  -> 
( A  e.  dom  G  /\  Fun  ( G  |`  { A } ) ) )
43ancoms 439 . . . . 5  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( A  e.  dom  G  /\  Fun  ( G  |`  { A } ) ) )
5 df-dfat 27985 . . . . . 6  |-  ( G defAt 
A  <->  ( A  e. 
dom  G  /\  Fun  ( G  |`  { A }
) ) )
6 afvfundmfveq 28012 . . . . . 6  |-  ( G defAt 
A  ->  ( G''' A )  =  ( G `
 A ) )
75, 6sylbir 204 . . . . 5  |-  ( ( A  e.  dom  G  /\  Fun  ( G  |`  { A } ) )  ->  ( G''' A )  =  ( G `  A ) )
84, 7syl 15 . . . 4  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( G''' A )  =  ( G `  A ) )
98eqcomd 2290 . . 3  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( G `  A
)  =  ( G''' A ) )
109eleq1d 2351 . 2  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( ( G `  A )  e.  dom  F  <-> 
( G''' A )  e.  dom  F ) )
111, 10bitrd 244 1  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( A  e.  dom  ( F  o.  G
)  <->  ( G''' A )  e.  dom  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1625    e. wcel 1686   {csn 3642   dom cdm 4691    |` cres 4693    o. ccom 4695   Fun wfun 5251   ` cfv 5257   defAt wdfat 27982  '''cafv 27983
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-fv 5265  df-dfat 27985  df-afv 27986
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