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Theorem fcompt 5365
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 5332 . . 3  |-  ( ( B : C --> D  /\  x  e.  C )  ->  ( B `  x
)  e.  D )
21adantll 460 . 2  |-  ( ( ( A : D --> E  /\  B : C --> D )  /\  x  e.  C )  ->  ( B `  x )  e.  D )
3 ffn 5077 . . . 4  |-  ( B : C --> D  ->  B  Fn  C )
43adantl 271 . . 3  |-  ( ( A : D --> E  /\  B : C --> D )  ->  B  Fn  C
)
5 dffn5im 5251 . . 3  |-  ( B  Fn  C  ->  B  =  ( x  e.  C  |->  ( B `  x ) ) )
64, 5syl 14 . 2  |-  ( ( A : D --> E  /\  B : C --> D )  ->  B  =  ( x  e.  C  |->  ( B `  x ) ) )
7 ffn 5077 . . . 4  |-  ( A : D --> E  ->  A  Fn  D )
87adantr 270 . . 3  |-  ( ( A : D --> E  /\  B : C --> D )  ->  A  Fn  D
)
9 dffn5im 5251 . . 3  |-  ( A  Fn  D  ->  A  =  ( y  e.  D  |->  ( A `  y ) ) )
108, 9syl 14 . 2  |-  ( ( A : D --> E  /\  B : C --> D )  ->  A  =  ( y  e.  D  |->  ( A `  y ) ) )
11 fveq2 5209 . 2  |-  ( y  =  ( B `  x )  ->  ( A `  y )  =  ( A `  ( B `  x ) ) )
122, 6, 10, 11fmptco 5362 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 102    = wceq 1285    e. wcel 1434    |-> cmpt 3847    o. ccom 4375    Fn wfn 4927   -->wf 4928   ` cfv 4932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fv 4940
This theorem is referenced by: (None)
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