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Theorem fcompt 5905
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.
StepHypRef Expression
1 ffvelrn 5869 . . 3  |-  ( ( B : C --> D  /\  x  e.  C )  ->  ( B `  x
)  e.  D )
21adantll 696 . 2  |-  ( ( ( A : D --> E  /\  B : C --> D )  /\  x  e.  C )  ->  ( B `  x )  e.  D )
3 ffn 5592 . . . 4  |-  ( B : C --> D  ->  B  Fn  C )
43adantl 454 . . 3  |-  ( ( A : D --> E  /\  B : C --> D )  ->  B  Fn  C
)
5 dffn5 5773 . . 3  |-  ( B  Fn  C  <->  B  =  ( x  e.  C  |->  ( B `  x
) ) )
64, 5sylib 190 . 2  |-  ( ( A : D --> E  /\  B : C --> D )  ->  B  =  ( x  e.  C  |->  ( B `  x ) ) )
7 ffn 5592 . . . 4  |-  ( A : D --> E  ->  A  Fn  D )
87adantr 453 . . 3  |-  ( ( A : D --> E  /\  B : C --> D )  ->  A  Fn  D
)
9 dffn5 5773 . . 3  |-  ( A  Fn  D  <->  A  =  ( y  e.  D  |->  ( A `  y
) ) )
108, 9sylib 190 . 2  |-  ( ( A : D --> E  /\  B : C --> D )  ->  A  =  ( y  e.  D  |->  ( A `  y ) ) )
11 fveq2 5729 . 2  |-  ( y  =  ( B `  x )  ->  ( A `  y )  =  ( A `  ( B `  x ) ) )
122, 6, 10, 11fmptco 5902 1  |-  ( ( A : D --> E  /\  B : C --> D )  ->  ( A  o.  B )  =  ( x  e.  C  |->  ( A `  ( B `
 x ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    e. cmpt 4267    o. ccom 4883    Fn wfn 5450   -->wf 5451   ` cfv 5455
This theorem is referenced by:  revco  11804  caucvgrlem2  12469  fucidcl  14163  fucsect  14170  prf1st  14302  prf2nd  14303  curfcl  14330  yonedalem4c  14375  yonedalem3b  14377  yonedainv  14379  frmdup3  14812  efginvrel1  15361  frgpup3lem  15410  frgpup3  15411  dprdfinv  15578  prdstps  17662  imasdsf1olem  18404  meascnbl  24574  gamcvg2lem  24844  mzprename  26807  grpvlinv  27428  grpvrinv  27429  mhmvlin  27430  mendassa  27480  mulc1cncfg  27698  expcnfg  27703
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fv 5463
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