Theorem fcompt 5813

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 5776	. . 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 5479	. . . 4 |- ( B : C --> D -> B Fn C )
4	3	adantl 277	. . 3 |- ( ( A : D --> E /\ B : C --> D ) -> B Fn C )
5		dffn5im 5687	. . 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 5479	. . . 4 |- ( A : D --> E -> A Fn D )
8	7	adantr 276	. . 3 |- ( ( A : D --> E /\ B : C --> D ) -> A Fn D )
9		dffn5im 5687	. . 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 5635	. 2 |- ( y = ( B ` x ) -> ( A ` y ) = ( A ` ( B ` x ) ) )
12	2, 6, 10, 11	fmptco 5809	1 |- ( ( A : D --> E /\ B : C --> D ) -> ( A o. B ) = ( x e. C |-> ( A ` ( B ` x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   |-> cmpt 4148   o. ccom 4727   Fn wfn 5319   -->wf 5320   ` cfv 5324

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332

This theorem is referenced by: (None)