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Theorem hof2val 14046
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 14045 . 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 5889 . . . 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 5892 . . 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 4122 . 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 5899 . . . 4  |-  ( X H Y )  e. 
_V
1918mptex 5762 . . 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 5991 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 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   2ndc2nd 6137   Basecbs 13164    Hom chom 13235  compcco 13236   Catccat 13582  HomFchof 14038
This theorem is referenced by:  hof2  14047  hofcllem  14048  hofcl  14049  yonedalem3b  14069
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-hof 14040
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