Theorem fcompt

Description: Express composition of two functions as a maps-to applying both in sequence. (Contributed by Stefan O'Rear, 5-Oct-2014) (Proof shortened by Mario Carneiro, 27-Dec-2014)

	Ref	Expression
Assertion	fcompt	|- ( ( A : D --> E /\ B : C --> D ) -> ( A o. B ) = ( x e. C |-> ( A ` ( B ` x ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ffvelrn	|- ( ( B : C --> D /\ x e. C ) -> ( B ` x ) e. D )
2	1	adantll	|- ( ( ( A : D --> E /\ B : C --> D ) /\ x e. C ) -> ( B ` x ) e. D )
3		ffn	|- ( B : C --> D -> B Fn C )
4	3	adantl	|- ( ( A : D --> E /\ B : C --> D ) -> B Fn C )
5		dffn5	|- ( B Fn C <-> B = ( x e. C |-> ( B ` x ) ) )
6	4 5	sylib	|- ( ( A : D --> E /\ B : C --> D ) -> B = ( x e. C |-> ( B ` x ) ) )
7		ffn	|- ( A : D --> E -> A Fn D )
8	7	adantr	|- ( ( A : D --> E /\ B : C --> D ) -> A Fn D )
9		dffn5	|- ( A Fn D <-> A = ( y e. D |-> ( A ` y ) ) )
10	8 9	sylib	|- ( ( A : D --> E /\ B : C --> D ) -> A = ( y e. D |-> ( A ` y ) ) )
11		fveq2	|- ( y = ( B ` x ) -> ( A ` y ) = ( A ` ( B ` x ) ) )
12	2 6 10 11	fmptco	|- ( ( A : D --> E /\ B : C --> D ) -> ( A o. B ) = ( x e. C |-> ( A ` ( B ` x ) ) ) )