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Theorem hof2 16084
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 16083 . 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 462 . . . 4  |-  ( (
ph  /\  h  =  K )  ->  h  =  K )
1413oveq2d 6321 . . 3  |-  ( (
ph  /\  h  =  K )  ->  ( G ( <. X ,  Y >.  .x.  W )
h )  =  ( G ( <. X ,  Y >.  .x.  W ) K ) )
1514oveq1d 6320 . 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 6333 . . 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 5970 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 setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087   <.cop 4008   ` cfv 5601  (class class class)co 6305   2ndc2nd 6806   Basecbs 15075   Hom chom 15154  compcco 15155   Catccat 15512  HomFchof 16075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-hof 16077
This theorem is referenced by:  yon12  16092  yon2  16093
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