Metamath Proof Explorer

Theorem dmfcoafv

Description: Domains of a function composition, analogous to dmfco . (Contributed by Alexander van der Vekens, 23-Jul-2017)

Ref Expression
Assertion dmfcoafv 
|- ( ( Fun G /\ A e. dom G ) -> ( A e. dom ( F o. G ) <-> ( G ''' A ) e. dom F ) )

Proof

Step Hyp Ref Expression
1 dmfco 
 |-  ( ( Fun G /\ A e. dom G ) -> ( A e. dom ( F o. G ) <-> ( G ` A ) e. dom F ) )
2 funres 
 |-  ( Fun G -> Fun ( G |` { A } ) )
3 2 anim2i 
 |-  ( ( A e. dom G /\ Fun G ) -> ( A e. dom G /\ Fun ( G |` { A } ) ) )
4 3 ancoms 
 |-  ( ( Fun G /\ A e. dom G ) -> ( A e. dom G /\ Fun ( G |` { A } ) ) )
5 df-dfat 
 |-  ( G defAt A <-> ( A e. dom G /\ Fun ( G |` { A } ) ) )
6 afvfundmfveq 
 |-  ( G defAt A -> ( G ''' A ) = ( G ` A ) )
7 5 6 sylbir 
 |-  ( ( A e. dom G /\ Fun ( G |` { A } ) ) -> ( G ''' A ) = ( G ` A ) )
8 4 7 syl 
 |-  ( ( Fun G /\ A e. dom G ) -> ( G ''' A ) = ( G ` A ) )
9 8 eqcomd 
 |-  ( ( Fun G /\ A e. dom G ) -> ( G ` A ) = ( G ''' A ) )
10 9 eleq1d 
 |-  ( ( Fun G /\ A e. dom G ) -> ( ( G ` A ) e. dom F <-> ( G ''' A ) e. dom F ) )
11 1 10 bitrd 
 |-  ( ( Fun G /\ A e. dom G ) -> ( A e. dom ( F o. G ) <-> ( G ''' A ) e. dom F ) )