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Theorem fcompt 6074
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 6035 . . 3  |-  ( ( B : C --> D  /\  x  e.  C )  ->  ( B `  x
)  e.  D )
21adantll 718 . 2  |-  ( ( ( A : D --> E  /\  B : C --> D )  /\  x  e.  C )  ->  ( B `  x )  e.  D )
3 ffn 5746 . . . 4  |-  ( B : C --> D  ->  B  Fn  C )
43adantl 467 . . 3  |-  ( ( A : D --> E  /\  B : C --> D )  ->  B  Fn  C
)
5 dffn5 5926 . . 3  |-  ( B  Fn  C  <->  B  =  ( x  e.  C  |->  ( B `  x
) ) )
64, 5sylib 199 . 2  |-  ( ( A : D --> E  /\  B : C --> D )  ->  B  =  ( x  e.  C  |->  ( B `  x ) ) )
7 ffn 5746 . . . 4  |-  ( A : D --> E  ->  A  Fn  D )
87adantr 466 . . 3  |-  ( ( A : D --> E  /\  B : C --> D )  ->  A  Fn  D
)
9 dffn5 5926 . . 3  |-  ( A  Fn  D  <->  A  =  ( y  e.  D  |->  ( A `  y
) ) )
108, 9sylib 199 . 2  |-  ( ( A : D --> E  /\  B : C --> D )  ->  A  =  ( y  e.  D  |->  ( A `  y ) ) )
11 fveq2 5881 . 2  |-  ( y  =  ( B `  x )  ->  ( A `  y )  =  ( A `  ( B `  x ) ) )
122, 6, 10, 11fmptco 6071 1  |-  ( ( A : D --> E  /\  B : C --> D )  ->  ( A  o.  B )  =  ( x  e.  C  |->  ( A `  ( B `
 x ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870    |-> cmpt 4484    o. ccom 4858    Fn wfn 5596   -->wf 5597   ` cfv 5601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609
This theorem is referenced by:  2fvcoidd  6210  revco  12916  repsco  12921  caucvgrlem2  13718  fucidcl  15821  fucsect  15828  prf1st  16040  prf2nd  16041  curfcl  16068  yonedalem4c  16113  yonedalem3b  16115  yonedainv  16117  frmdup3  16602  efginvrel1  17313  frgpup3lem  17362  frgpup3  17363  dprdfinv  17587  grpvlinv  19351  grpvrinv  19352  mhmvlin  19353  chcoeffeqlem  19840  prdstps  20575  imasdsf1olem  21319  gamcvg2lem  23849  meascnbl  28880  elmrsubrn  29946  mzprename  35300  mendassa  35759  mulc1cncfg  37239  expcnfg  37243  cncficcgt0  37338  dvsinax  37355  dirkercncflem2  37535  fourierdlem18  37556  fourierdlem53  37591  fourierdlem93  37631  fourierdlem101  37639  fourierdlem111  37649  sge0resrnlem  37779  omeiunle  37847
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