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Theorem hof2 15178
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 15177 . 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 461 . . . 4  |-  ( (
ph  /\  h  =  K )  ->  h  =  K )
1413oveq2d 6209 . . 3  |-  ( (
ph  /\  h  =  K )  ->  ( G ( <. X ,  Y >.  .x.  W )
h )  =  ( G ( <. X ,  Y >.  .x.  W ) K ) )
1514oveq1d 6208 . 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 6218 . . 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 5881 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 369    = wceq 1370    e. wcel 1758   _Vcvv 3071   <.cop 3984   ` cfv 5519  (class class class)co 6193   2ndc2nd 6679   Basecbs 14285   Hom chom 14360  compcco 14361   Catccat 14713  HomFchof 15169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681  df-hof 15171
This theorem is referenced by:  yon12  15186  yon2  15187
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