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Theorem dmfcoafv 32050
Description: Domains of a function composition, analogous to dmfco 5948. (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 5948 . 2  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( A  e.  dom  ( F  o.  G
)  <->  ( G `  A )  e.  dom  F ) )
2 funres 5633 . . . . . . 7  |-  ( Fun 
G  ->  Fun  ( G  |`  { A } ) )
32anim2i 569 . . . . . 6  |-  ( ( A  e.  dom  G  /\  Fun  G )  -> 
( A  e.  dom  G  /\  Fun  ( G  |`  { A } ) ) )
43ancoms 453 . . . . 5  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( A  e.  dom  G  /\  Fun  ( G  |`  { A } ) ) )
5 df-dfat 31991 . . . . . 6  |-  ( G defAt 
A  <->  ( A  e. 
dom  G  /\  Fun  ( G  |`  { A }
) ) )
6 afvfundmfveq 32013 . . . . . 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 2475 . . 3  |-  ( ( Fun  G  /\  A  e.  dom  G )  -> 
( G `  A
)  =  ( G''' A ) )
109eleq1d 2536 . 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 1379    e. wcel 1767   {csn 4033   dom cdm 5005    |` cres 5007    o. ccom 5009   Fun wfun 5588   ` cfv 5594   defAt wdfat 31988  '''cafv 31989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-res 5017  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-dfat 31991  df-afv 31992
This theorem is referenced by: (None)
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