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Theorem cofuval 13772
Description: Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
cofuval.b  |-  B  =  ( Base `  C
)
cofuval.f  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
cofuval.g  |-  ( ph  ->  G  e.  ( D 
Func  E ) )
Assertion
Ref Expression
cofuval  |-  ( ph  ->  ( G  o.func  F )  =  <. ( ( 1st `  G )  o.  ( 1st `  F ) ) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) ) >. )
Distinct variable groups:    x, y, B    x, F, y    x, G, y    ph, x, y
Allowed substitution hints:    C( x, y)    D( x, y)    E( x, y)

Proof of Theorem cofuval
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cofu 13750 . . 3  |-  o.func  =  (
g  e.  _V , 
f  e.  _V  |->  <.
( ( 1st `  g
)  o.  ( 1st `  f ) ) ,  ( x  e.  dom  dom  ( 2nd `  f
) ,  y  e. 
dom  dom  ( 2nd `  f
)  |->  ( ( ( ( 1st `  f
) `  x )
( 2nd `  g
) ( ( 1st `  f ) `  y
) )  o.  (
x ( 2nd `  f
) y ) ) ) >. )
21a1i 10 . 2  |-  ( ph  ->  o.func  =  ( g  e. 
_V ,  f  e. 
_V  |->  <. ( ( 1st `  g )  o.  ( 1st `  f ) ) ,  ( x  e. 
dom  dom  ( 2nd `  f
) ,  y  e. 
dom  dom  ( 2nd `  f
)  |->  ( ( ( ( 1st `  f
) `  x )
( 2nd `  g
) ( ( 1st `  f ) `  y
) )  o.  (
x ( 2nd `  f
) y ) ) ) >. ) )
3 simprl 732 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
g  =  G )
43fveq2d 5545 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 1st `  g
)  =  ( 1st `  G ) )
5 simprr 733 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
f  =  F )
65fveq2d 5545 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 1st `  f
)  =  ( 1st `  F ) )
74, 6coeq12d 4864 . . 3  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( ( 1st `  g
)  o.  ( 1st `  f ) )  =  ( ( 1st `  G
)  o.  ( 1st `  F ) ) )
85fveq2d 5545 . . . . . . . 8  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 2nd `  f
)  =  ( 2nd `  F ) )
98dmeqd 4897 . . . . . . 7  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  dom  ( 2nd `  f
)  =  dom  ( 2nd `  F ) )
10 cofuval.b . . . . . . . . . 10  |-  B  =  ( Base `  C
)
11 relfunc 13752 . . . . . . . . . . 11  |-  Rel  ( C  Func  D )
12 cofuval.f . . . . . . . . . . 11  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
13 1st2ndbr 6185 . . . . . . . . . . 11  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
1411, 12, 13sylancr 644 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
1510, 14funcfn2 13759 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  F
)  Fn  ( B  X.  B ) )
16 fndm 5359 . . . . . . . . 9  |-  ( ( 2nd `  F )  Fn  ( B  X.  B )  ->  dom  ( 2nd `  F )  =  ( B  X.  B ) )
1715, 16syl 15 . . . . . . . 8  |-  ( ph  ->  dom  ( 2nd `  F
)  =  ( B  X.  B ) )
1817adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  dom  ( 2nd `  F
)  =  ( B  X.  B ) )
199, 18eqtrd 2328 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  dom  ( 2nd `  f
)  =  ( B  X.  B ) )
2019dmeqd 4897 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  dom  dom  ( 2nd `  f
)  =  dom  ( B  X.  B ) )
21 dmxpid 4914 . . . . 5  |-  dom  ( B  X.  B )  =  B
2220, 21syl6eq 2344 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  dom  dom  ( 2nd `  f
)  =  B )
233fveq2d 5545 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( 2nd `  g
)  =  ( 2nd `  G ) )
246fveq1d 5543 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( ( 1st `  f
) `  x )  =  ( ( 1st `  F ) `  x
) )
256fveq1d 5543 . . . . . 6  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( ( 1st `  f
) `  y )  =  ( ( 1st `  F ) `  y
) )
2623, 24, 25oveq123d 5895 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( ( ( 1st `  f ) `  x
) ( 2nd `  g
) ( ( 1st `  f ) `  y
) )  =  ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) ) )
278oveqd 5891 . . . . 5  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( x ( 2nd `  f ) y )  =  ( x ( 2nd `  F ) y ) )
2826, 27coeq12d 4864 . . . 4  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( ( ( ( 1st `  f ) `
 x ) ( 2nd `  g ) ( ( 1st `  f
) `  y )
)  o.  ( x ( 2nd `  f
) y ) )  =  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) )
2922, 22, 28mpt2eq123dv 5926 . . 3  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  -> 
( x  e.  dom  dom  ( 2nd `  f
) ,  y  e. 
dom  dom  ( 2nd `  f
)  |->  ( ( ( ( 1st `  f
) `  x )
( 2nd `  g
) ( ( 1st `  f ) `  y
) )  o.  (
x ( 2nd `  f
) y ) ) )  =  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) ) )
307, 29opeq12d 3820 . 2  |-  ( (
ph  /\  ( g  =  G  /\  f  =  F ) )  ->  <. ( ( 1st `  g
)  o.  ( 1st `  f ) ) ,  ( x  e.  dom  dom  ( 2nd `  f
) ,  y  e. 
dom  dom  ( 2nd `  f
)  |->  ( ( ( ( 1st `  f
) `  x )
( 2nd `  g
) ( ( 1st `  f ) `  y
) )  o.  (
x ( 2nd `  f
) y ) ) ) >.  =  <. ( ( 1st `  G
)  o.  ( 1st `  F ) ) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F ) `
 x ) ( 2nd `  G ) ( ( 1st `  F
) `  y )
)  o.  ( x ( 2nd `  F
) y ) ) ) >. )
31 cofuval.g . . 3  |-  ( ph  ->  G  e.  ( D 
Func  E ) )
32 elex 2809 . . 3  |-  ( G  e.  ( D  Func  E )  ->  G  e.  _V )
3331, 32syl 15 . 2  |-  ( ph  ->  G  e.  _V )
34 elex 2809 . . 3  |-  ( F  e.  ( C  Func  D )  ->  F  e.  _V )
3512, 34syl 15 . 2  |-  ( ph  ->  F  e.  _V )
36 opex 4253 . . 3  |-  <. (
( 1st `  G
)  o.  ( 1st `  F ) ) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F ) `
 x ) ( 2nd `  G ) ( ( 1st `  F
) `  y )
)  o.  ( x ( 2nd `  F
) y ) ) ) >.  e.  _V
3736a1i 10 . 2  |-  ( ph  -> 
<. ( ( 1st `  G
)  o.  ( 1st `  F ) ) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F ) `
 x ) ( 2nd `  G ) ( ( 1st `  F
) `  y )
)  o.  ( x ( 2nd `  F
) y ) ) ) >.  e.  _V )
382, 30, 33, 35, 37ovmpt2d 5991 1  |-  ( ph  ->  ( G  o.func  F )  =  <. ( ( 1st `  G )  o.  ( 1st `  F ) ) ,  ( x  e.  B ,  y  e.  B  |->  ( ( ( ( 1st `  F
) `  x )
( 2nd `  G
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) ) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   class class class wbr 4039    X. cxp 4703   dom cdm 4705    o. ccom 4709   Rel wrel 4710    Fn wfn 5266   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137   Basecbs 13164    Func cfunc 13744    o.func ccofu 13746
This theorem is referenced by:  cofu1st  13773  cofu2nd  13775  cofuval2  13777  cofucl  13778  cofuass  13779  cofulid  13780  cofurid  13781  prf1st  13994  prf2nd  13995
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-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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-reu 2563  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-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  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-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-ixp 6834  df-func 13748  df-cofu 13750
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