Theorem fcompt 5852

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 5815	. . 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 5513	. . . 4 |- ( B : C --> D -> B Fn C )
4	3	adantl 277	. . 3 |- ( ( A : D --> E /\ B : C --> D ) -> B Fn C )
5		dffn5im 5727	. . 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 5513	. . . 4 |- ( A : D --> E -> A Fn D )
8	7	adantr 276	. . 3 |- ( ( A : D --> E /\ B : C --> D ) -> A Fn D )
9		dffn5im 5727	. . 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 5675	. 2 |- ( y = ( B ` x ) -> ( A ` y ) = ( A ` ( B ` x ) ) )
12	2, 6, 10, 11	fmptco 5848	1 |- ( ( A : D --> E /\ B : C --> D ) -> ( A o. B ) = ( x e. C |-> ( A ` ( B ` x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   |-> cmpt 4176   o. ccom 4758   Fn wfn 5352   -->wf 5353   ` cfv 5357

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365

This theorem is referenced by: (None)