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Theorem fucval 14155
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 14141 . . . 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 11 . . 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 733 . . . . . . 7  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
t  =  C )
5 simprr 734 . . . . . . 7  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  ->  u  =  D )
64, 5oveq12d 6099 . . . . . 6  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( t  Func  u
)  =  ( C 
Func  D ) )
7 fucval.b . . . . . 6  |-  B  =  ( C  Func  D
)
86, 7syl6eqr 2486 . . . . 5  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( t  Func  u
)  =  B )
98opeq2d 3991 . . . 4  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  ->  <. ( Base `  ndx ) ,  ( t  Func  u ) >.  =  <. (
Base `  ndx ) ,  B >. )
104, 5oveq12d 6099 . . . . . 6  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( t Nat  u )  =  ( C Nat  D
) )
11 fucval.n . . . . . 6  |-  N  =  ( C Nat  D )
1210, 11syl6eqr 2486 . . . . 5  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( t Nat  u )  =  N )
1312opeq2d 3991 . . . 4  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  ->  <. (  Hom  `  ndx ) ,  ( t Nat  u ) >.  =  <. (  Hom  `  ndx ) ,  N >. )
148, 8xpeq12d 4903 . . . . . . 7  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( ( t  Func  u )  X.  ( t 
Func  u ) )  =  ( B  X.  B
) )
1512oveqd 6098 . . . . . . . . . 10  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( g ( t Nat  u ) h )  =  ( g N h ) )
1612oveqd 6098 . . . . . . . . . 10  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( f ( t Nat  u ) g )  =  ( f N g ) )
174fveq2d 5732 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( Base `  t )  =  ( Base `  C
) )
18 fucval.a . . . . . . . . . . . 12  |-  A  =  ( Base `  C
)
1917, 18syl6eqr 2486 . . . . . . . . . . 11  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( Base `  t )  =  A )
205fveq2d 5732 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
(comp `  u )  =  (comp `  D )
)
21 fucval.o . . . . . . . . . . . . . 14  |-  .x.  =  (comp `  D )
2220, 21syl6eqr 2486 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
(comp `  u )  =  .x.  )
2322oveqd 6098 . . . . . . . . . . . 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 6098 . . . . . . . . . . 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 4287 . . . . . . . . . 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 6136 . . . . . . . . 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 3276 . . . . . . . 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 3276 . . . . . . 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 6136 . . . . . 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 452 . . . . . 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 2471 . . . . 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 3991 . . . 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 3898 . . 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 4708 . . . 4  |-  { <. (
Base `  ndx ) ,  B >. ,  <. (  Hom  `  ndx ) ,  N >. ,  <. (comp ` 
ndx ) ,  .xb  >. }  e.  _V
3837a1i 11 . . 3  |-  ( ph  ->  { <. ( Base `  ndx ) ,  B >. , 
<. (  Hom  `  ndx ) ,  N >. , 
<. (comp `  ndx ) , 
.xb  >. }  e.  _V )
393, 34, 35, 36, 38ovmpt2d 6201 . 2  |-  ( ph  ->  ( C FuncCat  D )  =  { <. ( Base `  ndx ) ,  B >. , 
<. (  Hom  `  ndx ) ,  N >. , 
<. (comp `  ndx ) , 
.xb  >. } )
401, 39syl5eq 2480 1  |-  ( ph  ->  Q  =  { <. (
Base `  ndx ) ,  B >. ,  <. (  Hom  `  ndx ) ,  N >. ,  <. (comp ` 
ndx ) ,  .xb  >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   [_csb 3251   {ctp 3816   <.cop 3817    e. cmpt 4266    X. cxp 4876   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   1stc1st 6347   2ndc2nd 6348   ndxcnx 13466   Basecbs 13469    Hom chom 13540  compcco 13541   Catccat 13889    Func cfunc 14051   Nat cnat 14138   FuncCat cfuc 14139
This theorem is referenced by:  fuccofval  14156  fucbas  14157  fuchom  14158  fucpropd  14174  catcfuccl  14264
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-fuc 14141
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