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Theorem hofval 14026
Description: Value of the Hom functor, which is a bifunctor (a functor of two arguments), contravariant in the first argument and covariant in the second, from  (oppCat `  C )  X.  C to  SetCat, whose object part is the hom-function 
Hom, and with morphism part given by pre- and post-composition. (Contributed by Mario Carneiro, 15-Jan-2017.)
Hypotheses
Ref Expression
hofval.m  |-  M  =  (HomF
`  C )
hofval.c  |-  ( ph  ->  C  e.  Cat )
hofval.b  |-  B  =  ( Base `  C
)
hofval.h  |-  H  =  (  Hom  `  C
)
hofval.o  |-  .x.  =  (comp `  C )
Assertion
Ref Expression
hofval  |-  ( ph  ->  M  =  <. (  Homf  `  C ) ,  ( x  e.  ( B  X.  B ) ,  y  e.  ( B  X.  B )  |->  ( f  e.  ( ( 1st `  y ) H ( 1st `  x
) ) ,  g  e.  ( ( 2nd `  x ) H ( 2nd `  y ) )  |->  ( h  e.  ( H `  x
)  |->  ( ( g ( x  .x.  ( 2nd `  y ) ) h ) ( <.
( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) ) ) ) >. )
Distinct variable groups:    f, g, h, x, y, B    ph, f,
g, h, x, y    C, f, g, h, x, y    f, H, g, h, x, y    .x. , f,
g, h, x, y
Allowed substitution hints:    M( x, y, f, g, h)

Proof of Theorem hofval
Dummy variables  b 
c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hofval.m . 2  |-  M  =  (HomF
`  C )
2 df-hof 14024 . . . 4  |- HomF  =  ( c  e.  Cat  |->  <. (  Homf  `  c ) ,  [_ ( Base `  c )  /  b ]_ ( x  e.  ( b  X.  b ) ,  y  e.  ( b  X.  b ) 
|->  ( f  e.  ( ( 1st `  y
) (  Hom  `  c
) ( 1st `  x
) ) ,  g  e.  ( ( 2nd `  x ) (  Hom  `  c ) ( 2nd `  y ) )  |->  ( h  e.  ( (  Hom  `  c ) `  x )  |->  ( ( g ( x (comp `  c ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x
) >. (comp `  c
) ( 2nd `  y
) ) f ) ) ) ) >.
)
32a1i 10 . . 3  |-  ( ph  -> HomF  =  ( c  e.  Cat  |->  <. (  Homf 
`  c ) , 
[_ ( Base `  c
)  /  b ]_ ( x  e.  (
b  X.  b ) ,  y  e.  ( b  X.  b ) 
|->  ( f  e.  ( ( 1st `  y
) (  Hom  `  c
) ( 1st `  x
) ) ,  g  e.  ( ( 2nd `  x ) (  Hom  `  c ) ( 2nd `  y ) )  |->  ( h  e.  ( (  Hom  `  c ) `  x )  |->  ( ( g ( x (comp `  c ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x
) >. (comp `  c
) ( 2nd `  y
) ) f ) ) ) ) >.
) )
4 simpr 447 . . . . 5  |-  ( (
ph  /\  c  =  C )  ->  c  =  C )
54fveq2d 5529 . . . 4  |-  ( (
ph  /\  c  =  C )  ->  (  Homf  `  c )  =  (  Homf 
`  C ) )
6 fvex 5539 . . . . . 6  |-  ( Base `  c )  e.  _V
76a1i 10 . . . . 5  |-  ( (
ph  /\  c  =  C )  ->  ( Base `  c )  e. 
_V )
84fveq2d 5529 . . . . . 6  |-  ( (
ph  /\  c  =  C )  ->  ( Base `  c )  =  ( Base `  C
) )
9 hofval.b . . . . . 6  |-  B  =  ( Base `  C
)
108, 9syl6eqr 2333 . . . . 5  |-  ( (
ph  /\  c  =  C )  ->  ( Base `  c )  =  B )
11 simpr 447 . . . . . . 7  |-  ( ( ( ph  /\  c  =  C )  /\  b  =  B )  ->  b  =  B )
1211, 11xpeq12d 4714 . . . . . 6  |-  ( ( ( ph  /\  c  =  C )  /\  b  =  B )  ->  (
b  X.  b )  =  ( B  X.  B ) )
13 simplr 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  c  =  C )  /\  b  =  B )  ->  c  =  C )
1413fveq2d 5529 . . . . . . . . 9  |-  ( ( ( ph  /\  c  =  C )  /\  b  =  B )  ->  (  Hom  `  c )  =  (  Hom  `  C
) )
15 hofval.h . . . . . . . . 9  |-  H  =  (  Hom  `  C
)
1614, 15syl6eqr 2333 . . . . . . . 8  |-  ( ( ( ph  /\  c  =  C )  /\  b  =  B )  ->  (  Hom  `  c )  =  H )
1716oveqd 5875 . . . . . . 7  |-  ( ( ( ph  /\  c  =  C )  /\  b  =  B )  ->  (
( 1st `  y
) (  Hom  `  c
) ( 1st `  x
) )  =  ( ( 1st `  y
) H ( 1st `  x ) ) )
1816oveqd 5875 . . . . . . 7  |-  ( ( ( ph  /\  c  =  C )  /\  b  =  B )  ->  (
( 2nd `  x
) (  Hom  `  c
) ( 2nd `  y
) )  =  ( ( 2nd `  x
) H ( 2nd `  y ) ) )
1916fveq1d 5527 . . . . . . . 8  |-  ( ( ( ph  /\  c  =  C )  /\  b  =  B )  ->  (
(  Hom  `  c ) `
 x )  =  ( H `  x
) )
2013fveq2d 5529 . . . . . . . . . . 11  |-  ( ( ( ph  /\  c  =  C )  /\  b  =  B )  ->  (comp `  c )  =  (comp `  C ) )
21 hofval.o . . . . . . . . . . 11  |-  .x.  =  (comp `  C )
2220, 21syl6eqr 2333 . . . . . . . . . 10  |-  ( ( ( ph  /\  c  =  C )  /\  b  =  B )  ->  (comp `  c )  =  .x.  )
2322oveqd 5875 . . . . . . . . 9  |-  ( ( ( ph  /\  c  =  C )  /\  b  =  B )  ->  ( <. ( 1st `  y
) ,  ( 1st `  x ) >. (comp `  c ) ( 2nd `  y ) )  =  ( <. ( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) )
2422oveqd 5875 . . . . . . . . . 10  |-  ( ( ( ph  /\  c  =  C )  /\  b  =  B )  ->  (
x (comp `  c
) ( 2nd `  y
) )  =  ( x  .x.  ( 2nd `  y ) ) )
2524oveqd 5875 . . . . . . . . 9  |-  ( ( ( ph  /\  c  =  C )  /\  b  =  B )  ->  (
g ( x (comp `  c ) ( 2nd `  y ) ) h )  =  ( g ( x  .x.  ( 2nd `  y ) ) h ) )
26 eqidd 2284 . . . . . . . . 9  |-  ( ( ( ph  /\  c  =  C )  /\  b  =  B )  ->  f  =  f )
2723, 25, 26oveq123d 5879 . . . . . . . 8  |-  ( ( ( ph  /\  c  =  C )  /\  b  =  B )  ->  (
( g ( x (comp `  c )
( 2nd `  y
) ) h ) ( <. ( 1st `  y
) ,  ( 1st `  x ) >. (comp `  c ) ( 2nd `  y ) ) f )  =  ( ( g ( x  .x.  ( 2nd `  y ) ) h ) (
<. ( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) )
2819, 27mpteq12dv 4098 . . . . . . 7  |-  ( ( ( ph  /\  c  =  C )  /\  b  =  B )  ->  (
h  e.  ( (  Hom  `  c ) `  x )  |->  ( ( g ( x (comp `  c ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x
) >. (comp `  c
) ( 2nd `  y
) ) f ) )  =  ( h  e.  ( H `  x )  |->  ( ( g ( x  .x.  ( 2nd `  y ) ) h ) (
<. ( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) ) )
2917, 18, 28mpt2eq123dv 5910 . . . . . 6  |-  ( ( ( ph  /\  c  =  C )  /\  b  =  B )  ->  (
f  e.  ( ( 1st `  y ) (  Hom  `  c
) ( 1st `  x
) ) ,  g  e.  ( ( 2nd `  x ) (  Hom  `  c ) ( 2nd `  y ) )  |->  ( h  e.  ( (  Hom  `  c ) `  x )  |->  ( ( g ( x (comp `  c ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x
) >. (comp `  c
) ( 2nd `  y
) ) f ) ) )  =  ( f  e.  ( ( 1st `  y ) H ( 1st `  x
) ) ,  g  e.  ( ( 2nd `  x ) H ( 2nd `  y ) )  |->  ( h  e.  ( H `  x
)  |->  ( ( g ( x  .x.  ( 2nd `  y ) ) h ) ( <.
( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) ) ) )
3012, 12, 29mpt2eq123dv 5910 . . . . 5  |-  ( ( ( ph  /\  c  =  C )  /\  b  =  B )  ->  (
x  e.  ( b  X.  b ) ,  y  e.  ( b  X.  b )  |->  ( f  e.  ( ( 1st `  y ) (  Hom  `  c
) ( 1st `  x
) ) ,  g  e.  ( ( 2nd `  x ) (  Hom  `  c ) ( 2nd `  y ) )  |->  ( h  e.  ( (  Hom  `  c ) `  x )  |->  ( ( g ( x (comp `  c ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x
) >. (comp `  c
) ( 2nd `  y
) ) f ) ) ) )  =  ( x  e.  ( B  X.  B ) ,  y  e.  ( B  X.  B ) 
|->  ( f  e.  ( ( 1st `  y
) H ( 1st `  x ) ) ,  g  e.  ( ( 2nd `  x ) H ( 2nd `  y
) )  |->  ( h  e.  ( H `  x )  |->  ( ( g ( x  .x.  ( 2nd `  y ) ) h ) (
<. ( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) ) ) ) )
317, 10, 30csbied2 3124 . . . 4  |-  ( (
ph  /\  c  =  C )  ->  [_ ( Base `  c )  / 
b ]_ ( x  e.  ( b  X.  b
) ,  y  e.  ( b  X.  b
)  |->  ( f  e.  ( ( 1st `  y
) (  Hom  `  c
) ( 1st `  x
) ) ,  g  e.  ( ( 2nd `  x ) (  Hom  `  c ) ( 2nd `  y ) )  |->  ( h  e.  ( (  Hom  `  c ) `  x )  |->  ( ( g ( x (comp `  c ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x
) >. (comp `  c
) ( 2nd `  y
) ) f ) ) ) )  =  ( x  e.  ( B  X.  B ) ,  y  e.  ( B  X.  B ) 
|->  ( f  e.  ( ( 1st `  y
) H ( 1st `  x ) ) ,  g  e.  ( ( 2nd `  x ) H ( 2nd `  y
) )  |->  ( h  e.  ( H `  x )  |->  ( ( g ( x  .x.  ( 2nd `  y ) ) h ) (
<. ( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) ) ) ) )
325, 31opeq12d 3804 . . 3  |-  ( (
ph  /\  c  =  C )  ->  <. (  Homf  `  c ) ,  [_ ( Base `  c )  /  b ]_ (
x  e.  ( b  X.  b ) ,  y  e.  ( b  X.  b )  |->  ( f  e.  ( ( 1st `  y ) (  Hom  `  c
) ( 1st `  x
) ) ,  g  e.  ( ( 2nd `  x ) (  Hom  `  c ) ( 2nd `  y ) )  |->  ( h  e.  ( (  Hom  `  c ) `  x )  |->  ( ( g ( x (comp `  c ) ( 2nd `  y ) ) h ) ( <. ( 1st `  y ) ,  ( 1st `  x
) >. (comp `  c
) ( 2nd `  y
) ) f ) ) ) ) >.  =  <. (  Homf  `  C ) ,  ( x  e.  ( B  X.  B
) ,  y  e.  ( B  X.  B
)  |->  ( f  e.  ( ( 1st `  y
) H ( 1st `  x ) ) ,  g  e.  ( ( 2nd `  x ) H ( 2nd `  y
) )  |->  ( h  e.  ( H `  x )  |->  ( ( g ( x  .x.  ( 2nd `  y ) ) h ) (
<. ( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) ) ) ) >. )
33 hofval.c . . 3  |-  ( ph  ->  C  e.  Cat )
34 opex 4237 . . . 4  |-  <. (  Homf  `  C ) ,  ( x  e.  ( B  X.  B ) ,  y  e.  ( B  X.  B )  |->  ( f  e.  ( ( 1st `  y ) H ( 1st `  x
) ) ,  g  e.  ( ( 2nd `  x ) H ( 2nd `  y ) )  |->  ( h  e.  ( H `  x
)  |->  ( ( g ( x  .x.  ( 2nd `  y ) ) h ) ( <.
( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) ) ) ) >.  e.  _V
3534a1i 10 . . 3  |-  ( ph  -> 
<. (  Homf 
`  C ) ,  ( x  e.  ( B  X.  B ) ,  y  e.  ( B  X.  B ) 
|->  ( f  e.  ( ( 1st `  y
) H ( 1st `  x ) ) ,  g  e.  ( ( 2nd `  x ) H ( 2nd `  y
) )  |->  ( h  e.  ( H `  x )  |->  ( ( g ( x  .x.  ( 2nd `  y ) ) h ) (
<. ( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) ) ) ) >.  e.  _V )
363, 32, 33, 35fvmptd 5606 . 2  |-  ( ph  ->  (HomF
`  C )  = 
<. (  Homf 
`  C ) ,  ( x  e.  ( B  X.  B ) ,  y  e.  ( B  X.  B ) 
|->  ( f  e.  ( ( 1st `  y
) H ( 1st `  x ) ) ,  g  e.  ( ( 2nd `  x ) H ( 2nd `  y
) )  |->  ( h  e.  ( H `  x )  |->  ( ( g ( x  .x.  ( 2nd `  y ) ) h ) (
<. ( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) ) ) ) >. )
371, 36syl5eq 2327 1  |-  ( ph  ->  M  =  <. (  Homf  `  C ) ,  ( x  e.  ( B  X.  B ) ,  y  e.  ( B  X.  B )  |->  ( f  e.  ( ( 1st `  y ) H ( 1st `  x
) ) ,  g  e.  ( ( 2nd `  x ) H ( 2nd `  y ) )  |->  ( h  e.  ( H `  x
)  |->  ( ( g ( x  .x.  ( 2nd `  y ) ) h ) ( <.
( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) ) ) ) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   [_csb 3081   <.cop 3643    e. cmpt 4077    X. cxp 4687   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566    Homf chomf 13568  HomFchof 14022
This theorem is referenced by:  hof1fval  14027  hof2fval  14029  hofcl  14033  hofpropd  14041
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-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
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-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-hof 14024
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