Theorem funcoeqres

Description: Express a constraint on a composition as a constraint on the composand. (Contributed by Stefan O'Rear, 7-Mar-2015)

	Ref	Expression
Assertion	funcoeqres	|- ( ( Fun G /\ ( F o. G ) = H ) -> ( F |` ran G ) = ( H o. `' G ) )

Proof

Step	Hyp	Ref	Expression
1		funcocnv2	|- ( Fun G -> ( G o. `' G ) = ( _I |` ran G ) )
2	1	coeq2d	|- ( Fun G -> ( F o. ( G o. `' G ) ) = ( F o. ( _I |` ran G ) ) )
3		coass	|- ( ( F o. G ) o. `' G ) = ( F o. ( G o. `' G ) )
4	3	eqcomi	|- ( F o. ( G o. `' G ) ) = ( ( F o. G ) o. `' G )
5		coires1	|- ( F o. ( _I |` ran G ) ) = ( F |` ran G )
6	2 4 5	3eqtr3g	|- ( Fun G -> ( ( F o. G ) o. `' G ) = ( F |` ran G ) )
7		coeq1	|- ( ( F o. G ) = H -> ( ( F o. G ) o. `' G ) = ( H o. `' G ) )
8	6 7	sylan9req	|- ( ( Fun G /\ ( F o. G ) = H ) -> ( F |` ran G ) = ( H o. `' G ) )