Theorem fcompt 6400

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		ffvelrn 6357	. . 3 |- ( ( B : C --> D /\ x e. C ) -> ( B ` x ) e. D )
2	1	adantll 750	. 2 |- ( ( ( A : D --> E /\ B : C --> D ) /\ x e. C ) -> ( B ` x ) e. D )
3		ffn 6045	. . . 4 |- ( B : C --> D -> B Fn C )
4	3	adantl 482	. . 3 |- ( ( A : D --> E /\ B : C --> D ) -> B Fn C )
5		dffn5 6241	. . 3 |- ( B Fn C <-> B = ( x e. C |-> ( B ` x ) ) )
6	4, 5	sylib 208	. 2 |- ( ( A : D --> E /\ B : C --> D ) -> B = ( x e. C |-> ( B ` x ) ) )
7		ffn 6045	. . . 4 |- ( A : D --> E -> A Fn D )
8	7	adantr 481	. . 3 |- ( ( A : D --> E /\ B : C --> D ) -> A Fn D )
9		dffn5 6241	. . 3 |- ( A Fn D <-> A = ( y e. D |-> ( A ` y ) ) )
10	8, 9	sylib 208	. 2 |- ( ( A : D --> E /\ B : C --> D ) -> A = ( y e. D |-> ( A ` y ) ) )
11		fveq2 6191	. 2 |- ( y = ( B ` x ) -> ( A ` y ) = ( A ` ( B ` x ) ) )
12	2, 6, 10, 11	fmptco 6396	1 |- ( ( A : D --> E /\ B : C --> D ) -> ( A o. B ) = ( x e. C |-> ( A ` ( B ` x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   |-> cmpt 4729   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by:  2fvcoidd  6552  revco  13580  repsco  13585  caucvgrlem2  14405  fucidcl  16625  fucsect  16632  prf1st  16844  prf2nd  16845  curfcl  16872  yonedalem4c  16917  yonedalem3b  16919  yonedainv  16921  frmdup3  17404  efginvrel1  18141  frgpup3lem  18190  frgpup3  18191  dprdfinv  18418  grpvlinv  20201  grpvrinv  20202  mhmvlin  20203  chcoeffeqlem  20690  prdstps  21432  imasdsf1olem  22178  gamcvg2lem  24785  meascnbl  30282  elmrsubrn  31417  mzprename  37312  mendassa  37764  fcomptss  39395  mulc1cncfg  39821  expcnfg  39823  cncficcgt0  40101  fprodsubrecnncnvlem  40121  fprodaddrecnncnvlem  40123  dvsinax  40127  dirkercncflem2  40321  fourierdlem18  40342  fourierdlem53  40376  fourierdlem93  40416  fourierdlem101  40424  fourierdlem111  40434  sge0resrnlem  40620  omeiunle  40731  ovolval3  40861  amgmwlem  42548