Description: Value of a function composition. (Contributed by Glauco Siliprandi, 26-Jun-2021)
Ref | Expression | ||
---|---|---|---|
Hypotheses | fvcod.g | |- ( ph -> Fun G ) |
|
fvcod.a | |- ( ph -> A e. dom G ) |
||
fvcod.h | |- H = ( F o. G ) |
||
Assertion | fvcod | |- ( ph -> ( H ` A ) = ( F ` ( G ` A ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvcod.g | |- ( ph -> Fun G ) |
|
2 | fvcod.a | |- ( ph -> A e. dom G ) |
|
3 | fvcod.h | |- H = ( F o. G ) |
|
4 | 3 | fveq1i | |- ( H ` A ) = ( ( F o. G ) ` A ) |
5 | 4 | a1i | |- ( ph -> ( H ` A ) = ( ( F o. G ) ` A ) ) |
6 | fvco | |- ( ( Fun G /\ A e. dom G ) -> ( ( F o. G ) ` A ) = ( F ` ( G ` A ) ) ) |
|
7 | 1 2 6 | syl2anc | |- ( ph -> ( ( F o. G ) ` A ) = ( F ` ( G ` A ) ) ) |
8 | 5 7 | eqtrd | |- ( ph -> ( H ` A ) = ( F ` ( G ` A ) ) ) |