Theorem fcompt 5688

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.)

Assertion

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

Distinct variable groups:   x, A   x, B   x, C   x, D   x, E

Proof of Theorem fcompt

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffvelcdm 5651	. . 3 |- ( ( B : C --> D /\ x e. C ) -> ( B ` x ) e. D )
2	1	adantll 476	. 2 |- ( ( ( A : D --> E /\ B : C --> D ) /\ x e. C ) -> ( B ` x ) e. D )
3		ffn 5367	. . . 4 |- ( B : C --> D -> B Fn C )
4	3	adantl 277	. . 3 |- ( ( A : D --> E /\ B : C --> D ) -> B Fn C )
5		dffn5im 5563	. . 3 |- ( B Fn C -> B = ( x e. C |-> ( B ` x ) ) )
6	4, 5	syl 14	. 2 |- ( ( A : D --> E /\ B : C --> D ) -> B = ( x e. C |-> ( B ` x ) ) )
7		ffn 5367	. . . 4 |- ( A : D --> E -> A Fn D )
8	7	adantr 276	. . 3 |- ( ( A : D --> E /\ B : C --> D ) -> A Fn D )
9		dffn5im 5563	. . 3 |- ( A Fn D -> A = ( y e. D |-> ( A ` y ) ) )
10	8, 9	syl 14	. 2 |- ( ( A : D --> E /\ B : C --> D ) -> A = ( y e. D |-> ( A ` y ) ) )
11		fveq2 5517	. 2 |- ( y = ( B ` x ) -> ( A ` y ) = ( A ` ( B ` x ) ) )
12	2, 6, 10, 11	fmptco 5684	1 |- ( ( A : D --> E /\ B : C --> D ) -> ( A o. B ) = ( x e. C |-> ( A ` ( B ` x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   |-> cmpt 4066   o. ccom 4632   Fn wfn 5213   -->wf 5214   ` cfv 5218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226

This theorem is referenced by: (None)