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Theorem fcompt 5710
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 5679 . . 3  |-  ( ( B : C --> D  /\  x  e.  C )  ->  ( B `  x
)  e.  D )
21adantll 694 . 2  |-  ( ( ( A : D --> E  /\  B : C --> D )  /\  x  e.  C )  ->  ( B `  x )  e.  D )
3 ffn 5405 . . . 4  |-  ( B : C --> D  ->  B  Fn  C )
43adantl 452 . . 3  |-  ( ( A : D --> E  /\  B : C --> D )  ->  B  Fn  C
)
5 dffn5 5584 . . 3  |-  ( B  Fn  C  <->  B  =  ( x  e.  C  |->  ( B `  x
) ) )
64, 5sylib 188 . 2  |-  ( ( A : D --> E  /\  B : C --> D )  ->  B  =  ( x  e.  C  |->  ( B `  x ) ) )
7 ffn 5405 . . . 4  |-  ( A : D --> E  ->  A  Fn  D )
87adantr 451 . . 3  |-  ( ( A : D --> E  /\  B : C --> D )  ->  A  Fn  D
)
9 dffn5 5584 . . 3  |-  ( A  Fn  D  <->  A  =  ( y  e.  D  |->  ( A `  y
) ) )
108, 9sylib 188 . 2  |-  ( ( A : D --> E  /\  B : C --> D )  ->  A  =  ( y  e.  D  |->  ( A `  y ) ) )
11 fveq2 5541 . 2  |-  ( y  =  ( B `  x )  ->  ( A `  y )  =  ( A `  ( B `  x ) ) )
122, 6, 10, 11fmptco 5707 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 358    = wceq 1632    e. wcel 1696    e. cmpt 4093    o. ccom 4709    Fn wfn 5266   -->wf 5267   ` cfv 5271
This theorem is referenced by:  revco  11505  caucvgrlem2  12163  fucidcl  13855  fucsect  13862  prf1st  13994  prf2nd  13995  curfcl  14022  yonedalem4c  14067  yonedalem3b  14069  yonedainv  14071  frmdup3  14504  efginvrel1  15053  frgpup3lem  15102  frgpup3  15103  dprdfinv  15270  prdstps  17339  imasdsf1olem  17953  meascnbl  23561  mzprename  26930  grpvlinv  27553  grpvrinv  27554  mhmvlin  27555  mendassa  27605  mulc1cncfg  27824  expcnfg  27829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279
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