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Theorem dmfcoafv 29927
Description: Domains of a function composition, analogous to dmfco 5753. (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 5753 . 2  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( A  e.  dom  ( F  o.  G
)  <->  ( G `  A )  e.  dom  F ) )
2 funres 5445 . . . . . . 7  |-  ( Fun 
G  ->  Fun  ( G  |`  { A } ) )
32anim2i 564 . . . . . 6  |-  ( ( A  e.  dom  G  /\  Fun  G )  -> 
( A  e.  dom  G  /\  Fun  ( G  |`  { A } ) ) )
43ancoms 450 . . . . 5  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( A  e.  dom  G  /\  Fun  ( G  |`  { A } ) ) )
5 df-dfat 29866 . . . . . 6  |-  ( G defAt 
A  <->  ( A  e. 
dom  G  /\  Fun  ( G  |`  { A }
) ) )
6 afvfundmfveq 29890 . . . . . 6  |-  ( G defAt 
A  ->  ( G''' A )  =  ( G `
 A ) )
75, 6sylbir 213 . . . . 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 2438 . . 3  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( G `  A
)  =  ( G''' A ) )
109eleq1d 2499 . 2  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( ( G `  A )  e.  dom  F  <-> 
( G''' A )  e.  dom  F ) )
111, 10bitrd 253 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 184    /\ wa 369    = wceq 1362    e. wcel 1755   {csn 3865   dom cdm 4827    |` cres 4829    o. ccom 4831   Fun wfun 5400   ` cfv 5406   defAt wdfat 29863  '''cafv 29864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-res 4839  df-iota 5369  df-fun 5408  df-fn 5409  df-fv 5414  df-dfat 29866  df-afv 29867
This theorem is referenced by: (None)
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