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Theorem hof2fval 14272
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 )
Assertion
Ref Expression
hof2fval  |-  ( 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 ) ) ) )
Distinct variable groups:    f, g, h, B    ph, f, g, h    C, f, g, h   
f, H, g, h   
f, W, g, h    .x. , f, g, h    f, X, g, h    f, Y, g, h    f, Z, g, h
Allowed substitution hints:    M( f, g, h)

Proof of Theorem hof2fval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hofval.m . . . 4  |-  M  =  (HomF
`  C )
2 hofval.c . . . 4  |-  ( ph  ->  C  e.  Cat )
3 hof1.b . . . 4  |-  B  =  ( Base `  C
)
4 hof1.h . . . 4  |-  H  =  (  Hom  `  C
)
5 hof2.o . . . 4  |-  .x.  =  (comp `  C )
61, 2, 3, 4, 5hofval 14269 . . 3  |-  ( ph  ->  M  =  <. (  Homf  `  C ) ,  ( x  e.  ( B  X.  B ) ,  y  e.  ( B  X.  B )  |->  ( f  e.  ( ( 1st `  y ) H ( 1st `  x
) ) ,  g  e.  ( ( 2nd `  x ) H ( 2nd `  y ) )  |->  ( h  e.  ( H `  x
)  |->  ( ( g ( x  .x.  ( 2nd `  y ) ) h ) ( <.
( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) ) ) ) >. )
7 fvex 5675 . . . 4  |-  (  Homf  `  C )  e.  _V
8 fvex 5675 . . . . . . 7  |-  ( Base `  C )  e.  _V
93, 8eqeltri 2450 . . . . . 6  |-  B  e. 
_V
109, 9xpex 4923 . . . . 5  |-  ( B  X.  B )  e. 
_V
1110, 10mpt2ex 6357 . . . 4  |-  ( x  e.  ( B  X.  B ) ,  y  e.  ( B  X.  B )  |->  ( f  e.  ( ( 1st `  y ) H ( 1st `  x ) ) ,  g  e.  ( ( 2nd `  x
) H ( 2nd `  y ) )  |->  ( h  e.  ( H `
 x )  |->  ( ( g ( x 
.x.  ( 2nd `  y
) ) h ) ( <. ( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) ) ) )  e.  _V
127, 11op2ndd 6290 . . 3  |-  ( M  =  <. (  Homf  `  C ) ,  ( x  e.  ( B  X.  B
) ,  y  e.  ( B  X.  B
)  |->  ( f  e.  ( ( 1st `  y
) H ( 1st `  x ) ) ,  g  e.  ( ( 2nd `  x ) H ( 2nd `  y
) )  |->  ( h  e.  ( H `  x )  |->  ( ( g ( x  .x.  ( 2nd `  y ) ) h ) (
<. ( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) ) ) ) >.  ->  ( 2nd `  M )  =  ( x  e.  ( B  X.  B ) ,  y  e.  ( B  X.  B )  |->  ( f  e.  ( ( 1st `  y ) H ( 1st `  x
) ) ,  g  e.  ( ( 2nd `  x ) H ( 2nd `  y ) )  |->  ( h  e.  ( H `  x
)  |->  ( ( g ( x  .x.  ( 2nd `  y ) ) h ) ( <.
( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) ) ) ) )
136, 12syl 16 . 2  |-  ( ph  ->  ( 2nd `  M
)  =  ( x  e.  ( B  X.  B ) ,  y  e.  ( B  X.  B )  |->  ( f  e.  ( ( 1st `  y ) H ( 1st `  x ) ) ,  g  e.  ( ( 2nd `  x
) H ( 2nd `  y ) )  |->  ( h  e.  ( H `
 x )  |->  ( ( g ( x 
.x.  ( 2nd `  y
) ) h ) ( <. ( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) ) ) ) )
14 simprr 734 . . . . . 6  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  y  =  <. Z ,  W >. )
1514fveq2d 5665 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( 1st `  y )  =  ( 1st `  <. Z ,  W >. ) )
16 hof2.z . . . . . . 7  |-  ( ph  ->  Z  e.  B )
17 hof2.w . . . . . . 7  |-  ( ph  ->  W  e.  B )
18 op1stg 6291 . . . . . . 7  |-  ( ( Z  e.  B  /\  W  e.  B )  ->  ( 1st `  <. Z ,  W >. )  =  Z )
1916, 17, 18syl2anc 643 . . . . . 6  |-  ( ph  ->  ( 1st `  <. Z ,  W >. )  =  Z )
2019adantr 452 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( 1st ` 
<. Z ,  W >. )  =  Z )
2115, 20eqtrd 2412 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( 1st `  y )  =  Z )
22 simprl 733 . . . . . 6  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  x  =  <. X ,  Y >. )
2322fveq2d 5665 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( 1st `  x )  =  ( 1st `  <. X ,  Y >. ) )
24 hof1.x . . . . . . 7  |-  ( ph  ->  X  e.  B )
25 hof1.y . . . . . . 7  |-  ( ph  ->  Y  e.  B )
26 op1stg 6291 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 1st `  <. X ,  Y >. )  =  X )
2724, 25, 26syl2anc 643 . . . . . 6  |-  ( ph  ->  ( 1st `  <. X ,  Y >. )  =  X )
2827adantr 452 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( 1st ` 
<. X ,  Y >. )  =  X )
2923, 28eqtrd 2412 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( 1st `  x )  =  X )
3021, 29oveq12d 6031 . . 3  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( ( 1st `  y ) H ( 1st `  x
) )  =  ( Z H X ) )
3122fveq2d 5665 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( 2nd `  x )  =  ( 2nd `  <. X ,  Y >. ) )
32 op2ndg 6292 . . . . . . 7  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
3324, 25, 32syl2anc 643 . . . . . 6  |-  ( ph  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
3433adantr 452 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( 2nd ` 
<. X ,  Y >. )  =  Y )
3531, 34eqtrd 2412 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( 2nd `  x )  =  Y )
3614fveq2d 5665 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( 2nd `  y )  =  ( 2nd `  <. Z ,  W >. ) )
37 op2ndg 6292 . . . . . . 7  |-  ( ( Z  e.  B  /\  W  e.  B )  ->  ( 2nd `  <. Z ,  W >. )  =  W )
3816, 17, 37syl2anc 643 . . . . . 6  |-  ( ph  ->  ( 2nd `  <. Z ,  W >. )  =  W )
3938adantr 452 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( 2nd ` 
<. Z ,  W >. )  =  W )
4036, 39eqtrd 2412 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( 2nd `  y )  =  W )
4135, 40oveq12d 6031 . . 3  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( ( 2nd `  x ) H ( 2nd `  y
) )  =  ( Y H W ) )
4222fveq2d 5665 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( H `  x )  =  ( H `  <. X ,  Y >. ) )
43 df-ov 6016 . . . . 5  |-  ( X H Y )  =  ( H `  <. X ,  Y >. )
4442, 43syl6eqr 2430 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( H `  x )  =  ( X H Y ) )
4521, 29opeq12d 3927 . . . . . 6  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  <. ( 1st `  y ) ,  ( 1st `  x )
>.  =  <. Z ,  X >. )
4645, 40oveq12d 6031 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( <. ( 1st `  y ) ,  ( 1st `  x
) >.  .x.  ( 2nd `  y ) )  =  ( <. Z ,  X >.  .x.  W ) )
4722, 40oveq12d 6031 . . . . . 6  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( x  .x.  ( 2nd `  y
) )  =  (
<. X ,  Y >.  .x. 
W ) )
4847oveqd 6030 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( g
( x  .x.  ( 2nd `  y ) ) h )  =  ( g ( <. X ,  Y >.  .x.  W )
h ) )
49 eqidd 2381 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  f  =  f )
5046, 48, 49oveq123d 6034 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( (
g ( x  .x.  ( 2nd `  y ) ) h ) (
<. ( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f )  =  ( ( g ( <. X ,  Y >.  .x. 
W ) h ) ( <. Z ,  X >.  .x.  W ) f ) )
5144, 50mpteq12dv 4221 . . 3  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( h  e.  ( H `  x
)  |->  ( ( g ( x  .x.  ( 2nd `  y ) ) h ) ( <.
( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) )  =  ( h  e.  ( X H Y ) 
|->  ( ( g (
<. X ,  Y >.  .x. 
W ) h ) ( <. Z ,  X >.  .x.  W ) f ) ) )
5230, 41, 51mpt2eq123dv 6068 . 2  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  <. Z ,  W >. )
)  ->  ( f  e.  ( ( 1st `  y
) H ( 1st `  x ) ) ,  g  e.  ( ( 2nd `  x ) H ( 2nd `  y
) )  |->  ( h  e.  ( H `  x )  |->  ( ( g ( x  .x.  ( 2nd `  y ) ) h ) (
<. ( 1st `  y
) ,  ( 1st `  x ) >.  .x.  ( 2nd `  y ) ) f ) ) )  =  ( 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 ) ) ) )
53 opelxpi 4843 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
5424, 25, 53syl2anc 643 . 2  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
55 opelxpi 4843 . . 3  |-  ( ( Z  e.  B  /\  W  e.  B )  -> 
<. Z ,  W >.  e.  ( B  X.  B
) )
5616, 17, 55syl2anc 643 . 2  |-  ( ph  -> 
<. Z ,  W >.  e.  ( B  X.  B
) )
57 ovex 6038 . . . 4  |-  ( Z H X )  e. 
_V
58 ovex 6038 . . . 4  |-  ( Y H W )  e. 
_V
5957, 58mpt2ex 6357 . . 3  |-  ( 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 ) ) )  e.  _V
6059a1i 11 . 2  |-  ( ph  ->  ( 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 ) ) )  e. 
_V )
6113, 52, 54, 56, 60ovmpt2d 6133 1  |-  ( 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 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2892   <.cop 3753    e. cmpt 4200    X. cxp 4809   ` cfv 5387  (class class class)co 6013    e. cmpt2 6015   1stc1st 6279   2ndc2nd 6280   Basecbs 13389    Hom chom 13460  compcco 13461   Catccat 13809    Homf chomf 13811  HomFchof 14265
This theorem is referenced by:  hof2val  14273
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-hof 14267
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