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Theorem fuccofval 14148
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 )
fuccofval.x  |-  .xb  =  (comp `  Q )
Assertion
Ref Expression
fuccofval  |-  ( 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 ) ) ) ) ) )
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 fuccofval
StepHypRef Expression
1 fucval.q . . . 4  |-  Q  =  ( C FuncCat  D )
2 fucval.b . . . 4  |-  B  =  ( C  Func  D
)
3 fucval.n . . . 4  |-  N  =  ( C Nat  D )
4 fucval.a . . . 4  |-  A  =  ( Base `  C
)
5 fucval.o . . . 4  |-  .x.  =  (comp `  D )
6 fucval.c . . . 4  |-  ( ph  ->  C  e.  Cat )
7 fucval.d . . . 4  |-  ( ph  ->  D  e.  Cat )
8 eqidd 2436 . . . 4  |-  ( ph  ->  ( 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 ) ) ) ) )  =  ( 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 ) ) ) ) ) )
91, 2, 3, 4, 5, 6, 7, 8fucval 14147 . . 3  |-  ( ph  ->  Q  =  { <. (
Base `  ndx ) ,  B >. ,  <. (  Hom  `  ndx ) ,  N >. ,  <. (comp ` 
ndx ) ,  ( 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 ) ) ) ) )
>. } )
109fveq2d 5724 . 2  |-  ( ph  ->  (comp `  Q )  =  (comp `  { <. ( Base `  ndx ) ,  B >. ,  <. (  Hom  `  ndx ) ,  N >. ,  <. (comp ` 
ndx ) ,  ( 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 ) ) ) ) )
>. } ) )
11 fuccofval.x . 2  |-  .xb  =  (comp `  Q )
12 ovex 6098 . . . . . 6  |-  ( C 
Func  D )  e.  _V
132, 12eqeltri 2505 . . . . 5  |-  B  e. 
_V
1413, 13xpex 4982 . . . 4  |-  ( B  X.  B )  e. 
_V
1514, 13mpt2ex 6417 . . 3  |-  ( 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 ) ) ) ) )  e.  _V
16 catstr 14146 . . . 4  |-  { <. (
Base `  ndx ) ,  B >. ,  <. (  Hom  `  ndx ) ,  N >. ,  <. (comp ` 
ndx ) ,  ( 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 ) ) ) ) )
>. } Struct  <. 1 , ; 1 5 >.
17 ccoid 13637 . . . 4  |- comp  = Slot  (comp ` 
ndx )
18 snsstp3 3943 . . . 4  |-  { <. (comp `  ndx ) ,  ( 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 ) ) ) ) )
>. }  C_  { <. ( Base `  ndx ) ,  B >. ,  <. (  Hom  `  ndx ) ,  N >. ,  <. (comp ` 
ndx ) ,  ( 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 ) ) ) ) )
>. }
1916, 17, 18strfv 13493 . . 3  |-  ( ( 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 ) ) ) ) )  e.  _V  ->  (
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 ) ) ) ) )  =  (comp `  { <. ( Base `  ndx ) ,  B >. , 
<. (  Hom  `  ndx ) ,  N >. , 
<. (comp `  ndx ) ,  ( 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 ) ) ) ) )
>. } ) )
2015, 19ax-mp 8 . 2  |-  ( 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 ) ) ) ) )  =  (comp `  { <. ( Base `  ndx ) ,  B >. , 
<. (  Hom  `  ndx ) ,  N >. , 
<. (comp `  ndx ) ,  ( 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 ) ) ) ) )
>. } )
2110, 11, 203eqtr4g 2492 1  |-  ( 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 ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948   [_csb 3243   {ctp 3808   <.cop 3809    e. cmpt 4258    X. cxp 4868   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1stc1st 6339   2ndc2nd 6340   1c1 8983   5c5 10044  ;cdc 10374   ndxcnx 13458   Basecbs 13461    Hom chom 13532  compcco 13533   Catccat 13881    Func cfunc 14043   Nat cnat 14130   FuncCat cfuc 14131
This theorem is referenced by:  fucbas  14149  fuchom  14150  fucco  14151
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-hom 13545  df-cco 13546  df-fuc 14133
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