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Theorem hof2val 14030
Description: The morphism part of the Hom functor, for morphisms  <. f ,  g >. : <. X ,  Y >. --> <. Z ,  W >. (which since the first argument is contravariant means morphisms  f : Z --> X and  g : Y --> W), yields a function (a morphism of  SetCat) mapping  h : X --> Y to  g  o.  h  o.  f : Z --> W. (Contributed by Mario Carneiro, 15-Jan-2017.)
Hypotheses
Ref Expression
hofval.m  |-  M  =  (HomF
`  C )
hofval.c  |-  ( ph  ->  C  e.  Cat )
hof1.b  |-  B  =  ( Base `  C
)
hof1.h  |-  H  =  (  Hom  `  C
)
hof1.x  |-  ( ph  ->  X  e.  B )
hof1.y  |-  ( ph  ->  Y  e.  B )
hof2.z  |-  ( ph  ->  Z  e.  B )
hof2.w  |-  ( ph  ->  W  e.  B )
hof2.o  |-  .x.  =  (comp `  C )
hof2.f  |-  ( ph  ->  F  e.  ( Z H X ) )
hof2.g  |-  ( ph  ->  G  e.  ( Y H W ) )
Assertion
Ref Expression
hof2val  |-  ( ph  ->  ( F ( <. X ,  Y >. ( 2nd `  M )
<. Z ,  W >. ) G )  =  ( h  e.  ( X H Y )  |->  ( ( G ( <. X ,  Y >.  .x. 
W ) h ) ( <. Z ,  X >.  .x.  W ) F ) ) )
Distinct variable groups:    B, h    h, F    h, G    ph, h    C, h    h, H    h, W    .x. , h    h, X    h, Y    h, Z
Allowed substitution hint:    M( h)

Proof of Theorem hof2val
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hofval.m . . 3  |-  M  =  (HomF
`  C )
2 hofval.c . . 3  |-  ( ph  ->  C  e.  Cat )
3 hof1.b . . 3  |-  B  =  ( Base `  C
)
4 hof1.h . . 3  |-  H  =  (  Hom  `  C
)
5 hof1.x . . 3  |-  ( ph  ->  X  e.  B )
6 hof1.y . . 3  |-  ( ph  ->  Y  e.  B )
7 hof2.z . . 3  |-  ( ph  ->  Z  e.  B )
8 hof2.w . . 3  |-  ( ph  ->  W  e.  B )
9 hof2.o . . 3  |-  .x.  =  (comp `  C )
101, 2, 3, 4, 5, 6, 7, 8, 9hof2fval 14029 . 2  |-  ( ph  ->  ( <. X ,  Y >. ( 2nd `  M
) <. Z ,  W >. )  =  ( f  e.  ( Z H X ) ,  g  e.  ( Y H W )  |->  ( h  e.  ( X H Y )  |->  ( ( g ( <. X ,  Y >.  .x.  W )
h ) ( <. Z ,  X >.  .x. 
W ) f ) ) ) )
11 simplrr 737 . . . . 5  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  h  e.  ( X H Y ) )  ->  g  =  G )
1211oveq1d 5873 . . . 4  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  h  e.  ( X H Y ) )  ->  ( g
( <. X ,  Y >.  .x.  W ) h )  =  ( G ( <. X ,  Y >.  .x.  W ) h ) )
13 simplrl 736 . . . 4  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  h  e.  ( X H Y ) )  ->  f  =  F )
1412, 13oveq12d 5876 . . 3  |-  ( ( ( ph  /\  (
f  =  F  /\  g  =  G )
)  /\  h  e.  ( X H Y ) )  ->  ( (
g ( <. X ,  Y >.  .x.  W )
h ) ( <. Z ,  X >.  .x. 
W ) f )  =  ( ( G ( <. X ,  Y >.  .x.  W ) h ) ( <. Z ,  X >.  .x.  W ) F ) )
1514mpteq2dva 4106 . 2  |-  ( (
ph  /\  ( f  =  F  /\  g  =  G ) )  -> 
( h  e.  ( X H Y ) 
|->  ( ( g (
<. X ,  Y >.  .x. 
W ) h ) ( <. Z ,  X >.  .x.  W ) f ) )  =  ( h  e.  ( X H Y )  |->  ( ( G ( <. X ,  Y >.  .x. 
W ) h ) ( <. Z ,  X >.  .x.  W ) F ) ) )
16 hof2.f . 2  |-  ( ph  ->  F  e.  ( Z H X ) )
17 hof2.g . 2  |-  ( ph  ->  G  e.  ( Y H W ) )
18 ovex 5883 . . . 4  |-  ( X H Y )  e. 
_V
1918mptex 5746 . . 3  |-  ( h  e.  ( X H Y )  |->  ( ( G ( <. X ,  Y >.  .x.  W )
h ) ( <. Z ,  X >.  .x. 
W ) F ) )  e.  _V
2019a1i 10 . 2  |-  ( ph  ->  ( h  e.  ( X H Y ) 
|->  ( ( G (
<. X ,  Y >.  .x. 
W ) h ) ( <. Z ,  X >.  .x.  W ) F ) )  e.  _V )
2110, 15, 16, 17, 20ovmpt2d 5975 1  |-  ( ph  ->  ( F ( <. X ,  Y >. ( 2nd `  M )
<. Z ,  W >. ) G )  =  ( h  e.  ( X H Y )  |->  ( ( G ( <. X ,  Y >.  .x. 
W ) h ) ( <. Z ,  X >.  .x.  W ) F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   2ndc2nd 6121   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566  HomFchof 14022
This theorem is referenced by:  hof2  14031  hofcllem  14032  hofcl  14033  yonedalem3b  14053
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  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-reu 2550  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-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  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-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-hof 14024
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