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Theorem fcompt 5694
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 5663 . . 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 5389 . . . 4  |-  ( B : C --> D  ->  B  Fn  C )
43adantl 452 . . 3  |-  ( ( A : D --> E  /\  B : C --> D )  ->  B  Fn  C
)
5 dffn5 5568 . . 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 5389 . . . 4  |-  ( A : D --> E  ->  A  Fn  D )
87adantr 451 . . 3  |-  ( ( A : D --> E  /\  B : C --> D )  ->  A  Fn  D
)
9 dffn5 5568 . . 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 5525 . 2  |-  ( y  =  ( B `  x )  ->  ( A `  y )  =  ( A `  ( B `  x ) ) )
122, 6, 10, 11fmptco 5691 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 1623    e. wcel 1684    e. cmpt 4077    o. ccom 4693    Fn wfn 5250   -->wf 5251   ` cfv 5255
This theorem is referenced by:  revco  11489  caucvgrlem2  12147  fucidcl  13839  fucsect  13846  prf1st  13978  prf2nd  13979  curfcl  14006  yonedalem4c  14051  yonedalem3b  14053  yonedainv  14055  frmdup3  14488  efginvrel1  15037  frgpup3lem  15086  frgpup3  15087  dprdfinv  15254  prdstps  17323  imasdsf1olem  17937  meascnbl  23546  mzprename  26827  grpvlinv  27450  grpvrinv  27451  mhmvlin  27452  mendassa  27502  mulc1cncfg  27721  expcnfg  27726
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263
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