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Theorem fucval 13832
Description: Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
fucval.q  |-  Q  =  ( C FuncCat  D )
fucval.b  |-  B  =  ( C  Func  D
)
fucval.n  |-  N  =  ( C Nat  D )
fucval.a  |-  A  =  ( Base `  C
)
fucval.o  |-  .x.  =  (comp `  D )
fucval.c  |-  ( ph  ->  C  e.  Cat )
fucval.d  |-  ( ph  ->  D  e.  Cat )
fucval.x  |-  ( ph  -> 
.xb  =  ( v  e.  ( B  X.  B ) ,  h  e.  B  |->  [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g N h ) ,  a  e.  ( f N g ) 
|->  ( x  e.  A  |->  ( ( b `  x ) ( <.
( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >.  .x.  ( ( 1st `  h ) `  x ) ) ( a `  x ) ) ) ) ) )
Assertion
Ref Expression
fucval  |-  ( ph  ->  Q  =  { <. (
Base `  ndx ) ,  B >. ,  <. (  Hom  `  ndx ) ,  N >. ,  <. (comp ` 
ndx ) ,  .xb  >. } )
Distinct variable groups:    v, h, B    a, b, f, g, h, v, x, ph    C, a, b, f, g, h, v, x    D, a, b, f, g, h, v, x
Allowed substitution hints:    A( x, v, f, g, h, a, b)    B( x, f, g, a, b)    Q( x, v, f, g, h, a, b)    .xb ( x, v, f, g, h, a, b)    .x. ( x, v, f, g, h, a, b)    N( x, v, f, g, h, a, b)

Proof of Theorem fucval
Dummy variables  t  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fucval.q . 2  |-  Q  =  ( C FuncCat  D )
2 df-fuc 13818 . . . 4  |- FuncCat  =  ( t  e.  Cat ,  u  e.  Cat  |->  { <. (
Base `  ndx ) ,  ( t  Func  u
) >. ,  <. (  Hom  `  ndx ) ,  ( t Nat  u )
>. ,  <. (comp `  ndx ) ,  ( v  e.  ( ( t 
Func  u )  X.  (
t  Func  u )
) ,  h  e.  ( t  Func  u
)  |->  [_ ( 1st `  v
)  /  f ]_ [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g ( t Nat  u
) h ) ,  a  e.  ( f ( t Nat  u ) g )  |->  ( x  e.  ( Base `  t
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  u
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) ) >. } )
32a1i 10 . . 3  |-  ( ph  -> FuncCat 
=  ( t  e. 
Cat ,  u  e.  Cat  |->  { <. ( Base `  ndx ) ,  ( t  Func  u
) >. ,  <. (  Hom  `  ndx ) ,  ( t Nat  u )
>. ,  <. (comp `  ndx ) ,  ( v  e.  ( ( t 
Func  u )  X.  (
t  Func  u )
) ,  h  e.  ( t  Func  u
)  |->  [_ ( 1st `  v
)  /  f ]_ [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g ( t Nat  u
) h ) ,  a  e.  ( f ( t Nat  u ) g )  |->  ( x  e.  ( Base `  t
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  u
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) ) >. } ) )
4 simprl 732 . . . . . . 7  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
t  =  C )
5 simprr 733 . . . . . . 7  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  ->  u  =  D )
64, 5oveq12d 5876 . . . . . 6  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( t  Func  u
)  =  ( C 
Func  D ) )
7 fucval.b . . . . . 6  |-  B  =  ( C  Func  D
)
86, 7syl6eqr 2333 . . . . 5  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( t  Func  u
)  =  B )
98opeq2d 3803 . . . 4  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  ->  <. ( Base `  ndx ) ,  ( t  Func  u ) >.  =  <. (
Base `  ndx ) ,  B >. )
104, 5oveq12d 5876 . . . . . 6  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( t Nat  u )  =  ( C Nat  D
) )
11 fucval.n . . . . . 6  |-  N  =  ( C Nat  D )
1210, 11syl6eqr 2333 . . . . 5  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( t Nat  u )  =  N )
1312opeq2d 3803 . . . 4  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  ->  <. (  Hom  `  ndx ) ,  ( t Nat  u ) >.  =  <. (  Hom  `  ndx ) ,  N >. )
148, 8xpeq12d 4714 . . . . . . 7  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( ( t  Func  u )  X.  ( t 
Func  u ) )  =  ( B  X.  B
) )
1512oveqd 5875 . . . . . . . . . 10  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( g ( t Nat  u ) h )  =  ( g N h ) )
1612oveqd 5875 . . . . . . . . . 10  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( f ( t Nat  u ) g )  =  ( f N g ) )
174fveq2d 5529 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( Base `  t )  =  ( Base `  C
) )
18 fucval.a . . . . . . . . . . . 12  |-  A  =  ( Base `  C
)
1917, 18syl6eqr 2333 . . . . . . . . . . 11  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( Base `  t )  =  A )
205fveq2d 5529 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
(comp `  u )  =  (comp `  D )
)
21 fucval.o . . . . . . . . . . . . . 14  |-  .x.  =  (comp `  D )
2220, 21syl6eqr 2333 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
(comp `  u )  =  .x.  )
2322oveqd 5875 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  u )
( ( 1st `  h
) `  x )
)  =  ( <.
( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >.  .x.  ( ( 1st `  h ) `  x ) ) )
2423oveqd 5875 . . . . . . . . . . 11  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( ( b `  x ) ( <.
( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  u
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) )  =  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >.  .x.  ( ( 1st `  h ) `  x ) ) ( a `  x ) ) )
2519, 24mpteq12dv 4098 . . . . . . . . . 10  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( x  e.  (
Base `  t )  |->  ( ( b `  x ) ( <.
( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  u
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) )  =  ( x  e.  A  |->  ( ( b `  x ) ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.  .x.  ( ( 1st `  h
) `  x )
) ( a `  x ) ) ) )
2615, 16, 25mpt2eq123dv 5910 . . . . . . . . 9  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( b  e.  ( g ( t Nat  u
) h ) ,  a  e.  ( f ( t Nat  u ) g )  |->  ( x  e.  ( Base `  t
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  u
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )  =  ( b  e.  ( g N h ) ,  a  e.  ( f N g )  |->  ( x  e.  A  |->  ( ( b `  x
) ( <. (
( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >.  .x.  ( ( 1st `  h ) `  x ) ) ( a `  x ) ) ) ) )
2726csbeq2dv 3106 . . . . . . . 8  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  ->  [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g ( t Nat  u
) h ) ,  a  e.  ( f ( t Nat  u ) g )  |->  ( x  e.  ( Base `  t
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  u
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )  =  [_ ( 2nd `  v )  /  g ]_ (
b  e.  ( g N h ) ,  a  e.  ( f N g )  |->  ( x  e.  A  |->  ( ( b `  x
) ( <. (
( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >.  .x.  ( ( 1st `  h ) `  x ) ) ( a `  x ) ) ) ) )
2827csbeq2dv 3106 . . . . . . 7  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  ->  [_ ( 1st `  v
)  /  f ]_ [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g ( t Nat  u
) h ) ,  a  e.  ( f ( t Nat  u ) g )  |->  ( x  e.  ( Base `  t
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  u
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )  =  [_ ( 1st `  v )  /  f ]_ [_ ( 2nd `  v )  / 
g ]_ ( b  e.  ( g N h ) ,  a  e.  ( f N g )  |->  ( x  e.  A  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >.  .x.  ( ( 1st `  h ) `  x ) ) ( a `  x ) ) ) ) )
2914, 8, 28mpt2eq123dv 5910 . . . . . 6  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( v  e.  ( ( t  Func  u
)  X.  ( t 
Func  u ) ) ,  h  e.  ( t 
Func  u )  |->  [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( t Nat  u
) h ) ,  a  e.  ( f ( t Nat  u ) g )  |->  ( x  e.  ( Base `  t
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  u
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) )  =  ( v  e.  ( B  X.  B ) ,  h  e.  B  |-> 
[_ ( 1st `  v
)  /  f ]_ [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g N h ) ,  a  e.  ( f N g ) 
|->  ( x  e.  A  |->  ( ( b `  x ) ( <.
( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >.  .x.  ( ( 1st `  h ) `  x ) ) ( a `  x ) ) ) ) ) )
30 fucval.x . . . . . . 7  |-  ( ph  -> 
.xb  =  ( v  e.  ( B  X.  B ) ,  h  e.  B  |->  [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g N h ) ,  a  e.  ( f N g ) 
|->  ( x  e.  A  |->  ( ( b `  x ) ( <.
( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >.  .x.  ( ( 1st `  h ) `  x ) ) ( a `  x ) ) ) ) ) )
3130adantr 451 . . . . . 6  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  ->  .xb  =  ( v  e.  ( B  X.  B
) ,  h  e.  B  |->  [_ ( 1st `  v
)  /  f ]_ [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g N h ) ,  a  e.  ( f N g ) 
|->  ( x  e.  A  |->  ( ( b `  x ) ( <.
( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >.  .x.  ( ( 1st `  h ) `  x ) ) ( a `  x ) ) ) ) ) )
3229, 31eqtr4d 2318 . . . . 5  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( v  e.  ( ( t  Func  u
)  X.  ( t 
Func  u ) ) ,  h  e.  ( t 
Func  u )  |->  [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( t Nat  u
) h ) ,  a  e.  ( f ( t Nat  u ) g )  |->  ( x  e.  ( Base `  t
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  u
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) )  = 
.xb  )
3332opeq2d 3803 . . . 4  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  ->  <. (comp `  ndx ) ,  ( v  e.  ( ( t  Func  u
)  X.  ( t 
Func  u ) ) ,  h  e.  ( t 
Func  u )  |->  [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( t Nat  u
) h ) ,  a  e.  ( f ( t Nat  u ) g )  |->  ( x  e.  ( Base `  t
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  u
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) ) >.  =  <. (comp `  ndx ) ,  .xb  >. )
349, 13, 33tpeq123d 3721 . . 3  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  ->  { <. ( Base `  ndx ) ,  ( t  Func  u ) >. ,  <. (  Hom  `  ndx ) ,  ( t Nat  u )
>. ,  <. (comp `  ndx ) ,  ( v  e.  ( ( t 
Func  u )  X.  (
t  Func  u )
) ,  h  e.  ( t  Func  u
)  |->  [_ ( 1st `  v
)  /  f ]_ [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g ( t Nat  u
) h ) ,  a  e.  ( f ( t Nat  u ) g )  |->  ( x  e.  ( Base `  t
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  u
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) ) >. }  =  { <. ( Base `  ndx ) ,  B >. ,  <. (  Hom  `  ndx ) ,  N >. ,  <. (comp ` 
ndx ) ,  .xb  >. } )
35 fucval.c . . 3  |-  ( ph  ->  C  e.  Cat )
36 fucval.d . . 3  |-  ( ph  ->  D  e.  Cat )
37 tpex 4519 . . . 4  |-  { <. (
Base `  ndx ) ,  B >. ,  <. (  Hom  `  ndx ) ,  N >. ,  <. (comp ` 
ndx ) ,  .xb  >. }  e.  _V
3837a1i 10 . . 3  |-  ( ph  ->  { <. ( Base `  ndx ) ,  B >. , 
<. (  Hom  `  ndx ) ,  N >. , 
<. (comp `  ndx ) , 
.xb  >. }  e.  _V )
393, 34, 35, 36, 38ovmpt2d 5975 . 2  |-  ( ph  ->  ( C FuncCat  D )  =  { <. ( Base `  ndx ) ,  B >. , 
<. (  Hom  `  ndx ) ,  N >. , 
<. (comp `  ndx ) , 
.xb  >. } )
401, 39syl5eq 2327 1  |-  ( ph  ->  Q  =  { <. (
Base `  ndx ) ,  B >. ,  <. (  Hom  `  ndx ) ,  N >. ,  <. (comp ` 
ndx ) ,  .xb  >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   [_csb 3081   {ctp 3642   <.cop 3643    e. cmpt 4077    X. cxp 4687   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   ndxcnx 13145   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566    Func cfunc 13728   Nat cnat 13815   FuncCat cfuc 13816
This theorem is referenced by:  fuccofval  13833  fucbas  13834  fuchom  13835  fucpropd  13851  catcfuccl  13941
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-pr 4214  ax-un 4512
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-tp 3648  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-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-fuc 13818
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