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Theorem fucval 14110
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 14096 . . . 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 6058 . . . . . 6  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( t  Func  u
)  =  ( C 
Func  D ) )
7 fucval.b . . . . . 6  |-  B  =  ( C  Func  D
)
86, 7syl6eqr 2454 . . . . 5  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( t  Func  u
)  =  B )
98opeq2d 3951 . . . 4  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  ->  <. ( Base `  ndx ) ,  ( t  Func  u ) >.  =  <. (
Base `  ndx ) ,  B >. )
104, 5oveq12d 6058 . . . . . 6  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( t Nat  u )  =  ( C Nat  D
) )
11 fucval.n . . . . . 6  |-  N  =  ( C Nat  D )
1210, 11syl6eqr 2454 . . . . 5  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( t Nat  u )  =  N )
1312opeq2d 3951 . . . 4  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  ->  <. (  Hom  `  ndx ) ,  ( t Nat  u ) >.  =  <. (  Hom  `  ndx ) ,  N >. )
148, 8xpeq12d 4862 . . . . . . 7  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( ( t  Func  u )  X.  ( t 
Func  u ) )  =  ( B  X.  B
) )
1512oveqd 6057 . . . . . . . . . 10  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( g ( t Nat  u ) h )  =  ( g N h ) )
1612oveqd 6057 . . . . . . . . . 10  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( f ( t Nat  u ) g )  =  ( f N g ) )
174fveq2d 5691 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( Base `  t )  =  ( Base `  C
) )
18 fucval.a . . . . . . . . . . . 12  |-  A  =  ( Base `  C
)
1917, 18syl6eqr 2454 . . . . . . . . . . 11  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
( Base `  t )  =  A )
205fveq2d 5691 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
(comp `  u )  =  (comp `  D )
)
21 fucval.o . . . . . . . . . . . . . 14  |-  .x.  =  (comp `  D )
2220, 21syl6eqr 2454 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( t  =  C  /\  u  =  D ) )  -> 
(comp `  u )  =  .x.  )
2322oveqd 6057 . . . . . . . . . . . 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 6057 . . . . . . . . . . 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 4247 . . . . . . . . . 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 6095 . . . . . . . . 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 3236 . . . . . . . 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 3236 . . . . . . 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 6095 . . . . . 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 2439 . . . . 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 3951 . . . 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 3858 . . 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 4667 . . . 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 6160 . 2  |-  ( ph  ->  ( C FuncCat  D )  =  { <. ( Base `  ndx ) ,  B >. , 
<. (  Hom  `  ndx ) ,  N >. , 
<. (comp `  ndx ) , 
.xb  >. } )
401, 39syl5eq 2448 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 1649    e. wcel 1721   _Vcvv 2916   [_csb 3211   {ctp 3776   <.cop 3777    e. cmpt 4226    X. cxp 4835   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1stc1st 6306   2ndc2nd 6307   ndxcnx 13421   Basecbs 13424    Hom chom 13495  compcco 13496   Catccat 13844    Func cfunc 14006   Nat cnat 14093   FuncCat cfuc 14094
This theorem is referenced by:  fuccofval  14111  fucbas  14112  fuchom  14113  fucpropd  14129  catcfuccl  14219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-fuc 14096
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