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Theorem dmfcoafv 27710
Description: Domains of a function composition, analogous to dmfco 5738. (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 5738 . 2  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( A  e.  dom  ( F  o.  G
)  <->  ( G `  A )  e.  dom  F ) )
2 funres 5434 . . . . . . 7  |-  ( Fun 
G  ->  Fun  ( G  |`  { A } ) )
32anim2i 553 . . . . . 6  |-  ( ( A  e.  dom  G  /\  Fun  G )  -> 
( A  e.  dom  G  /\  Fun  ( G  |`  { A } ) ) )
43ancoms 440 . . . . 5  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( A  e.  dom  G  /\  Fun  ( G  |`  { A } ) ) )
5 df-dfat 27644 . . . . . 6  |-  ( G defAt 
A  <->  ( A  e. 
dom  G  /\  Fun  ( G  |`  { A }
) ) )
6 afvfundmfveq 27673 . . . . . 6  |-  ( G defAt 
A  ->  ( G''' A )  =  ( G `
 A ) )
75, 6sylbir 205 . . . . 5  |-  ( ( A  e.  dom  G  /\  Fun  ( G  |`  { A } ) )  ->  ( G''' A )  =  ( G `  A ) )
84, 7syl 16 . . . 4  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( G''' A )  =  ( G `  A ) )
98eqcomd 2394 . . 3  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( G `  A
)  =  ( G''' A ) )
109eleq1d 2455 . 2  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( ( G `  A )  e.  dom  F  <-> 
( G''' A )  e.  dom  F ) )
111, 10bitrd 245 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   {csn 3759   dom cdm 4820    |` cres 4822    o. ccom 4824   Fun wfun 5390   ` cfv 5396   defAt wdfat 27641  '''cafv 27642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-res 4832  df-iota 5360  df-fun 5398  df-fn 5399  df-fv 5404  df-dfat 27644  df-afv 27645
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