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Theorem ofco 5966
Description: The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.)
Hypotheses
Ref Expression
ofco.1  |-  ( ph  ->  F  Fn  A )
ofco.2  |-  ( ph  ->  G  Fn  B )
ofco.3  |-  ( ph  ->  H : D --> C )
ofco.4  |-  ( ph  ->  A  e.  V )
ofco.5  |-  ( ph  ->  B  e.  W )
ofco.6  |-  ( ph  ->  D  e.  X )
ofco.7  |-  ( A  i^i  B )  =  C
Assertion
Ref Expression
ofco  |-  ( ph  ->  ( ( F  oF R G )  o.  H )  =  ( ( F  o.  H )  oF R ( G  o.  H ) ) )

Proof of Theorem ofco
Dummy variables  y  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ofco.3 . . . 4  |-  ( ph  ->  H : D --> C )
21ffvelrnda 5521 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  ( H `  x )  e.  C )
31feqmptd 5440 . . 3  |-  ( ph  ->  H  =  ( x  e.  D  |->  ( H `
 x ) ) )
4 ofco.1 . . . 4  |-  ( ph  ->  F  Fn  A )
5 ofco.2 . . . 4  |-  ( ph  ->  G  Fn  B )
6 ofco.4 . . . 4  |-  ( ph  ->  A  e.  V )
7 ofco.5 . . . 4  |-  ( ph  ->  B  e.  W )
8 ofco.7 . . . 4  |-  ( A  i^i  B )  =  C
9 eqidd 2116 . . . 4  |-  ( (
ph  /\  y  e.  A )  ->  ( F `  y )  =  ( F `  y ) )
10 eqidd 2116 . . . 4  |-  ( (
ph  /\  y  e.  B )  ->  ( G `  y )  =  ( G `  y ) )
114, 5, 6, 7, 8, 9, 10offval 5955 . . 3  |-  ( ph  ->  ( F  oF R G )  =  ( y  e.  C  |->  ( ( F `  y ) R ( G `  y ) ) ) )
12 fveq2 5387 . . . 4  |-  ( y  =  ( H `  x )  ->  ( F `  y )  =  ( F `  ( H `  x ) ) )
13 fveq2 5387 . . . 4  |-  ( y  =  ( H `  x )  ->  ( G `  y )  =  ( G `  ( H `  x ) ) )
1412, 13oveq12d 5758 . . 3  |-  ( y  =  ( H `  x )  ->  (
( F `  y
) R ( G `
 y ) )  =  ( ( F `
 ( H `  x ) ) R ( G `  ( H `  x )
) ) )
152, 3, 11, 14fmptco 5552 . 2  |-  ( ph  ->  ( ( F  oF R G )  o.  H )  =  ( x  e.  D  |->  ( ( F `  ( H `  x ) ) R ( G `
 ( H `  x ) ) ) ) )
16 inss1 3264 . . . . . 6  |-  ( A  i^i  B )  C_  A
178, 16eqsstrri 3098 . . . . 5  |-  C  C_  A
18 fss 5252 . . . . 5  |-  ( ( H : D --> C  /\  C  C_  A )  ->  H : D --> A )
191, 17, 18sylancl 407 . . . 4  |-  ( ph  ->  H : D --> A )
20 fnfco 5265 . . . 4  |-  ( ( F  Fn  A  /\  H : D --> A )  ->  ( F  o.  H )  Fn  D
)
214, 19, 20syl2anc 406 . . 3  |-  ( ph  ->  ( F  o.  H
)  Fn  D )
22 inss2 3265 . . . . . 6  |-  ( A  i^i  B )  C_  B
238, 22eqsstrri 3098 . . . . 5  |-  C  C_  B
24 fss 5252 . . . . 5  |-  ( ( H : D --> C  /\  C  C_  B )  ->  H : D --> B )
251, 23, 24sylancl 407 . . . 4  |-  ( ph  ->  H : D --> B )
26 fnfco 5265 . . . 4  |-  ( ( G  Fn  B  /\  H : D --> B )  ->  ( G  o.  H )  Fn  D
)
275, 25, 26syl2anc 406 . . 3  |-  ( ph  ->  ( G  o.  H
)  Fn  D )
28 ofco.6 . . 3  |-  ( ph  ->  D  e.  X )
29 inidm 3253 . . 3  |-  ( D  i^i  D )  =  D
30 ffn 5240 . . . . 5  |-  ( H : D --> C  ->  H  Fn  D )
311, 30syl 14 . . . 4  |-  ( ph  ->  H  Fn  D )
32 fvco2 5456 . . . 4  |-  ( ( H  Fn  D  /\  x  e.  D )  ->  ( ( F  o.  H ) `  x
)  =  ( F `
 ( H `  x ) ) )
3331, 32sylan 279 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  (
( F  o.  H
) `  x )  =  ( F `  ( H `  x ) ) )
34 fvco2 5456 . . . 4  |-  ( ( H  Fn  D  /\  x  e.  D )  ->  ( ( G  o.  H ) `  x
)  =  ( G `
 ( H `  x ) ) )
3531, 34sylan 279 . . 3  |-  ( (
ph  /\  x  e.  D )  ->  (
( G  o.  H
) `  x )  =  ( G `  ( H `  x ) ) )
3621, 27, 28, 28, 29, 33, 35offval 5955 . 2  |-  ( ph  ->  ( ( F  o.  H )  oF R ( G  o.  H ) )  =  ( x  e.  D  |->  ( ( F `  ( H `  x ) ) R ( G `
 ( H `  x ) ) ) ) )
3715, 36eqtr4d 2151 1  |-  ( ph  ->  ( ( F  oF R G )  o.  H )  =  ( ( F  o.  H )  oF R ( G  o.  H ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463    i^i cin 3038    C_ wss 3039    |-> cmpt 3957    o. ccom 4511    Fn wfn 5086   -->wf 5087   ` cfv 5091  (class class class)co 5740    oFcof 5946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-of 5948
This theorem is referenced by: (None)
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