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Theorem hof2 14359
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 ) )
hof2.k  |-  ( ph  ->  K  e.  ( X H Y ) )
Assertion
Ref Expression
hof2  |-  ( ph  ->  ( ( F (
<. X ,  Y >. ( 2nd `  M )
<. Z ,  W >. ) G ) `  K
)  =  ( ( G ( <. X ,  Y >.  .x.  W ) K ) ( <. Z ,  X >.  .x. 
W ) F ) )

Proof of Theorem hof2
Dummy variable  h is 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 )
10 hof2.f . . 3  |-  ( ph  ->  F  e.  ( Z H X ) )
11 hof2.g . . 3  |-  ( ph  ->  G  e.  ( Y H W ) )
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11hof2val 14358 . 2  |-  ( 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 ) ) )
13 simpr 449 . . . 4  |-  ( (
ph  /\  h  =  K )  ->  h  =  K )
1413oveq2d 6100 . . 3  |-  ( (
ph  /\  h  =  K )  ->  ( G ( <. X ,  Y >.  .x.  W )
h )  =  ( G ( <. X ,  Y >.  .x.  W ) K ) )
1514oveq1d 6099 . 2  |-  ( (
ph  /\  h  =  K )  ->  (
( G ( <. X ,  Y >.  .x. 
W ) h ) ( <. Z ,  X >.  .x.  W ) F )  =  ( ( G ( <. X ,  Y >.  .x.  W ) K ) ( <. Z ,  X >.  .x. 
W ) F ) )
16 hof2.k . 2  |-  ( ph  ->  K  e.  ( X H Y ) )
17 ovex 6109 . . 3  |-  ( ( G ( <. X ,  Y >.  .x.  W ) K ) ( <. Z ,  X >.  .x. 
W ) F )  e.  _V
1817a1i 11 . 2  |-  ( ph  ->  ( ( G (
<. X ,  Y >.  .x. 
W ) K ) ( <. Z ,  X >.  .x.  W ) F )  e.  _V )
1912, 15, 16, 18fvmptd 5813 1  |-  ( ph  ->  ( ( F (
<. X ,  Y >. ( 2nd `  M )
<. Z ,  W >. ) G ) `  K
)  =  ( ( G ( <. X ,  Y >.  .x.  W ) K ) ( <. Z ,  X >.  .x. 
W ) F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819   ` cfv 5457  (class class class)co 6084   2ndc2nd 6351   Basecbs 13474    Hom chom 13545  compcco 13546   Catccat 13894  HomFchof 14350
This theorem is referenced by:  yon12  14367  yon2  14368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-hof 14352
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