Theorem fcompt 5744

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 5707	. . 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 5419	. . . 4 |- ( B : C --> D -> B Fn C )
4	3	adantl 277	. . 3 |- ( ( A : D --> E /\ B : C --> D ) -> B Fn C )
5		dffn5im 5618	. . 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 5419	. . . 4 |- ( A : D --> E -> A Fn D )
8	7	adantr 276	. . 3 |- ( ( A : D --> E /\ B : C --> D ) -> A Fn D )
9		dffn5im 5618	. . 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 5570	. 2 |- ( y = ( B ` x ) -> ( A ` y ) = ( A ` ( B ` x ) ) )
12	2, 6, 10, 11	fmptco 5740	1 |- ( ( A : D --> E /\ B : C --> D ) -> ( A o. B ) = ( x e. C |-> ( A ` ( B ` x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   |-> cmpt 4104   o. ccom 4677   Fn wfn 5263   -->wf 5264   ` cfv 5268

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-fv 5276

This theorem is referenced by: (None)