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Theorem hofcllem 14048
Description: Lemma for hofcl 14049. (Contributed by Mario Carneiro, 15-Jan-2017.)
Hypotheses
Ref Expression
hofcl.m  |-  M  =  (HomF
`  C )
hofcl.o  |-  O  =  (oppCat `  C )
hofcl.d  |-  D  =  ( SetCat `  U )
hofcl.c  |-  ( ph  ->  C  e.  Cat )
hofcl.u  |-  ( ph  ->  U  e.  V )
hofcl.h  |-  ( ph  ->  ran  (  Homf  `  C ) 
C_  U )
hofcllem.b  |-  B  =  ( Base `  C
)
hofcllem.h  |-  H  =  (  Hom  `  C
)
hofcllem.x  |-  ( ph  ->  X  e.  B )
hofcllem.y  |-  ( ph  ->  Y  e.  B )
hofcllem.z  |-  ( ph  ->  Z  e.  B )
hofcllem.w  |-  ( ph  ->  W  e.  B )
hofcllem.s  |-  ( ph  ->  S  e.  B )
hofcllem.t  |-  ( ph  ->  T  e.  B )
hofcllem.m  |-  ( ph  ->  K  e.  ( Z H X ) )
hofcllem.n  |-  ( ph  ->  L  e.  ( Y H W ) )
hofcllem.p  |-  ( ph  ->  P  e.  ( S H Z ) )
hofcllem.q  |-  ( ph  ->  Q  e.  ( W H T ) )
Assertion
Ref Expression
hofcllem  |-  ( ph  ->  ( ( K (
<. S ,  Z >. (comp `  C ) X ) P ) ( <. X ,  Y >. ( 2nd `  M )
<. S ,  T >. ) ( Q ( <. Y ,  W >. (comp `  C ) T ) L ) )  =  ( ( P (
<. Z ,  W >. ( 2nd `  M )
<. S ,  T >. ) Q ) ( <.
( X H Y ) ,  ( Z H W ) >.
(comp `  D )
( S H T ) ) ( K ( <. X ,  Y >. ( 2nd `  M
) <. Z ,  W >. ) L ) ) )

Proof of Theorem hofcllem
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hofcllem.b . . . . 5  |-  B  =  ( Base `  C
)
2 hofcllem.h . . . . 5  |-  H  =  (  Hom  `  C
)
3 eqid 2296 . . . . 5  |-  (comp `  C )  =  (comp `  C )
4 hofcl.c . . . . . 6  |-  ( ph  ->  C  e.  Cat )
54adantr 451 . . . . 5  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  C  e.  Cat )
6 hofcllem.s . . . . . 6  |-  ( ph  ->  S  e.  B )
76adantr 451 . . . . 5  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  S  e.  B )
8 hofcllem.z . . . . . 6  |-  ( ph  ->  Z  e.  B )
98adantr 451 . . . . 5  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  Z  e.  B )
10 hofcllem.x . . . . . 6  |-  ( ph  ->  X  e.  B )
1110adantr 451 . . . . 5  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  X  e.  B )
12 hofcllem.p . . . . . 6  |-  ( ph  ->  P  e.  ( S H Z ) )
1312adantr 451 . . . . 5  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  P  e.  ( S H Z ) )
14 hofcllem.m . . . . . 6  |-  ( ph  ->  K  e.  ( Z H X ) )
1514adantr 451 . . . . 5  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  K  e.  ( Z H X ) )
16 hofcllem.t . . . . . 6  |-  ( ph  ->  T  e.  B )
1716adantr 451 . . . . 5  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  T  e.  B )
18 hofcllem.y . . . . . . 7  |-  ( ph  ->  Y  e.  B )
1918adantr 451 . . . . . 6  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  Y  e.  B )
20 simpr 447 . . . . . 6  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  f  e.  ( X H Y ) )
21 hofcllem.w . . . . . . . 8  |-  ( ph  ->  W  e.  B )
22 hofcllem.n . . . . . . . 8  |-  ( ph  ->  L  e.  ( Y H W ) )
23 hofcllem.q . . . . . . . 8  |-  ( ph  ->  Q  e.  ( W H T ) )
241, 2, 3, 4, 18, 21, 16, 22, 23catcocl 13603 . . . . . . 7  |-  ( ph  ->  ( Q ( <. Y ,  W >. (comp `  C ) T ) L )  e.  ( Y H T ) )
2524adantr 451 . . . . . 6  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  ( Q
( <. Y ,  W >. (comp `  C ) T ) L )  e.  ( Y H T ) )
261, 2, 3, 5, 11, 19, 17, 20, 25catcocl 13603 . . . . 5  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  ( ( Q ( <. Y ,  W >. (comp `  C
) T ) L ) ( <. X ,  Y >. (comp `  C
) T ) f )  e.  ( X H T ) )
271, 2, 3, 5, 7, 9, 11, 13, 15, 17, 26catass 13604 . . . 4  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  ( (
( ( Q (
<. Y ,  W >. (comp `  C ) T ) L ) ( <. X ,  Y >. (comp `  C ) T ) f ) ( <. Z ,  X >. (comp `  C ) T ) K ) ( <. S ,  Z >. (comp `  C ) T ) P )  =  ( ( ( Q (
<. Y ,  W >. (comp `  C ) T ) L ) ( <. X ,  Y >. (comp `  C ) T ) f ) ( <. S ,  X >. (comp `  C ) T ) ( K ( <. S ,  Z >. (comp `  C ) X ) P ) ) )
2821adantr 451 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  W  e.  B )
2922adantr 451 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  L  e.  ( Y H W ) )
3023adantr 451 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  Q  e.  ( W H T ) )
311, 2, 3, 5, 11, 19, 28, 20, 29, 17, 30catass 13604 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  ( ( Q ( <. Y ,  W >. (comp `  C
) T ) L ) ( <. X ,  Y >. (comp `  C
) T ) f )  =  ( Q ( <. X ,  W >. (comp `  C ) T ) ( L ( <. X ,  Y >. (comp `  C ) W ) f ) ) )
3231oveq1d 5889 . . . . . 6  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  ( (
( Q ( <. Y ,  W >. (comp `  C ) T ) L ) ( <. X ,  Y >. (comp `  C ) T ) f ) ( <. Z ,  X >. (comp `  C ) T ) K )  =  ( ( Q ( <. X ,  W >. (comp `  C ) T ) ( L ( <. X ,  Y >. (comp `  C ) W ) f ) ) (
<. Z ,  X >. (comp `  C ) T ) K ) )
331, 2, 3, 5, 11, 19, 28, 20, 29catcocl 13603 . . . . . . 7  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  ( L
( <. X ,  Y >. (comp `  C ) W ) f )  e.  ( X H W ) )
341, 2, 3, 5, 9, 11, 28, 15, 33, 17, 30catass 13604 . . . . . 6  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  ( ( Q ( <. X ,  W >. (comp `  C
) T ) ( L ( <. X ,  Y >. (comp `  C
) W ) f ) ) ( <. Z ,  X >. (comp `  C ) T ) K )  =  ( Q ( <. Z ,  W >. (comp `  C
) T ) ( ( L ( <. X ,  Y >. (comp `  C ) W ) f ) ( <. Z ,  X >. (comp `  C ) W ) K ) ) )
3532, 34eqtrd 2328 . . . . 5  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  ( (
( Q ( <. Y ,  W >. (comp `  C ) T ) L ) ( <. X ,  Y >. (comp `  C ) T ) f ) ( <. Z ,  X >. (comp `  C ) T ) K )  =  ( Q ( <. Z ,  W >. (comp `  C
) T ) ( ( L ( <. X ,  Y >. (comp `  C ) W ) f ) ( <. Z ,  X >. (comp `  C ) W ) K ) ) )
3635oveq1d 5889 . . . 4  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  ( (
( ( Q (
<. Y ,  W >. (comp `  C ) T ) L ) ( <. X ,  Y >. (comp `  C ) T ) f ) ( <. Z ,  X >. (comp `  C ) T ) K ) ( <. S ,  Z >. (comp `  C ) T ) P )  =  ( ( Q ( <. Z ,  W >. (comp `  C ) T ) ( ( L (
<. X ,  Y >. (comp `  C ) W ) f ) ( <. Z ,  X >. (comp `  C ) W ) K ) ) (
<. S ,  Z >. (comp `  C ) T ) P ) )
3727, 36eqtr3d 2330 . . 3  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  ( (
( Q ( <. Y ,  W >. (comp `  C ) T ) L ) ( <. X ,  Y >. (comp `  C ) T ) f ) ( <. S ,  X >. (comp `  C ) T ) ( K ( <. S ,  Z >. (comp `  C ) X ) P ) )  =  ( ( Q (
<. Z ,  W >. (comp `  C ) T ) ( ( L (
<. X ,  Y >. (comp `  C ) W ) f ) ( <. Z ,  X >. (comp `  C ) W ) K ) ) (
<. S ,  Z >. (comp `  C ) T ) P ) )
3837mpteq2dva 4122 . 2  |-  ( ph  ->  ( f  e.  ( X H Y ) 
|->  ( ( ( Q ( <. Y ,  W >. (comp `  C ) T ) L ) ( <. X ,  Y >. (comp `  C ) T ) f ) ( <. S ,  X >. (comp `  C ) T ) ( K ( <. S ,  Z >. (comp `  C ) X ) P ) ) )  =  ( f  e.  ( X H Y )  |->  ( ( Q ( <. Z ,  W >. (comp `  C ) T ) ( ( L (
<. X ,  Y >. (comp `  C ) W ) f ) ( <. Z ,  X >. (comp `  C ) W ) K ) ) (
<. S ,  Z >. (comp `  C ) T ) P ) ) )
39 hofcl.m . . 3  |-  M  =  (HomF
`  C )
401, 2, 3, 4, 6, 8, 10, 12, 14catcocl 13603 . . 3  |-  ( ph  ->  ( K ( <. S ,  Z >. (comp `  C ) X ) P )  e.  ( S H X ) )
4139, 4, 1, 2, 10, 18, 6, 16, 3, 40, 24hof2val 14046 . 2  |-  ( ph  ->  ( ( K (
<. S ,  Z >. (comp `  C ) X ) P ) ( <. X ,  Y >. ( 2nd `  M )
<. S ,  T >. ) ( Q ( <. Y ,  W >. (comp `  C ) T ) L ) )  =  ( f  e.  ( X H Y ) 
|->  ( ( ( Q ( <. Y ,  W >. (comp `  C ) T ) L ) ( <. X ,  Y >. (comp `  C ) T ) f ) ( <. S ,  X >. (comp `  C ) T ) ( K ( <. S ,  Z >. (comp `  C ) X ) P ) ) ) )
4239, 4, 1, 2, 8, 21, 6, 16, 3, 12, 23hof2val 14046 . . . 4  |-  ( ph  ->  ( P ( <. Z ,  W >. ( 2nd `  M )
<. S ,  T >. ) Q )  =  ( g  e.  ( Z H W )  |->  ( ( Q ( <. Z ,  W >. (comp `  C ) T ) g ) ( <. S ,  Z >. (comp `  C ) T ) P ) ) )
4339, 4, 1, 2, 10, 18, 8, 21, 3, 14, 22hof2val 14046 . . . 4  |-  ( ph  ->  ( K ( <. X ,  Y >. ( 2nd `  M )
<. Z ,  W >. ) L )  =  ( f  e.  ( X H Y )  |->  ( ( L ( <. X ,  Y >. (comp `  C ) W ) f ) ( <. Z ,  X >. (comp `  C ) W ) K ) ) )
4442, 43oveq12d 5892 . . 3  |-  ( ph  ->  ( ( P (
<. Z ,  W >. ( 2nd `  M )
<. S ,  T >. ) Q ) ( <.
( X H Y ) ,  ( Z H W ) >.
(comp `  D )
( S H T ) ) ( K ( <. X ,  Y >. ( 2nd `  M
) <. Z ,  W >. ) L ) )  =  ( ( g  e.  ( Z H W )  |->  ( ( Q ( <. Z ,  W >. (comp `  C
) T ) g ) ( <. S ,  Z >. (comp `  C
) T ) P ) ) ( <.
( X H Y ) ,  ( Z H W ) >.
(comp `  D )
( S H T ) ) ( f  e.  ( X H Y )  |->  ( ( L ( <. X ,  Y >. (comp `  C
) W ) f ) ( <. Z ,  X >. (comp `  C
) W ) K ) ) ) )
45 hofcl.d . . . 4  |-  D  =  ( SetCat `  U )
46 hofcl.u . . . 4  |-  ( ph  ->  U  e.  V )
47 eqid 2296 . . . 4  |-  (comp `  D )  =  (comp `  D )
48 eqid 2296 . . . . . 6  |-  (  Homf  `  C )  =  (  Homf 
`  C )
4948, 1, 2, 10, 18homfval 13611 . . . . 5  |-  ( ph  ->  ( X (  Homf  `  C ) Y )  =  ( X H Y ) )
5048, 1homffn 13612 . . . . . . . 8  |-  (  Homf  `  C )  Fn  ( B  X.  B )
5150a1i 10 . . . . . . 7  |-  ( ph  ->  (  Homf 
`  C )  Fn  ( B  X.  B
) )
52 hofcl.h . . . . . . 7  |-  ( ph  ->  ran  (  Homf  `  C ) 
C_  U )
53 df-f 5275 . . . . . . 7  |-  ( (  Homf 
`  C ) : ( B  X.  B
) --> U  <->  ( (  Homf  `  C )  Fn  ( B  X.  B )  /\  ran  (  Homf 
`  C )  C_  U ) )
5451, 52, 53sylanbrc 645 . . . . . 6  |-  ( ph  ->  (  Homf 
`  C ) : ( B  X.  B
) --> U )
5554, 10, 18fovrnd 6008 . . . . 5  |-  ( ph  ->  ( X (  Homf  `  C ) Y )  e.  U )
5649, 55eqeltrrd 2371 . . . 4  |-  ( ph  ->  ( X H Y )  e.  U )
5748, 1, 2, 8, 21homfval 13611 . . . . 5  |-  ( ph  ->  ( Z (  Homf  `  C ) W )  =  ( Z H W ) )
5854, 8, 21fovrnd 6008 . . . . 5  |-  ( ph  ->  ( Z (  Homf  `  C ) W )  e.  U )
5957, 58eqeltrrd 2371 . . . 4  |-  ( ph  ->  ( Z H W )  e.  U )
6048, 1, 2, 6, 16homfval 13611 . . . . 5  |-  ( ph  ->  ( S (  Homf  `  C ) T )  =  ( S H T ) )
6154, 6, 16fovrnd 6008 . . . . 5  |-  ( ph  ->  ( S (  Homf  `  C ) T )  e.  U )
6260, 61eqeltrrd 2371 . . . 4  |-  ( ph  ->  ( S H T )  e.  U )
631, 2, 3, 5, 9, 11, 28, 15, 33catcocl 13603 . . . . 5  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  ( ( L ( <. X ,  Y >. (comp `  C
) W ) f ) ( <. Z ,  X >. (comp `  C
) W ) K )  e.  ( Z H W ) )
64 eqid 2296 . . . . 5  |-  ( f  e.  ( X H Y )  |->  ( ( L ( <. X ,  Y >. (comp `  C
) W ) f ) ( <. Z ,  X >. (comp `  C
) W ) K ) )  =  ( f  e.  ( X H Y )  |->  ( ( L ( <. X ,  Y >. (comp `  C ) W ) f ) ( <. Z ,  X >. (comp `  C ) W ) K ) )
6563, 64fmptd 5700 . . . 4  |-  ( ph  ->  ( f  e.  ( X H Y ) 
|->  ( ( L (
<. X ,  Y >. (comp `  C ) W ) f ) ( <. Z ,  X >. (comp `  C ) W ) K ) ) : ( X H Y ) --> ( Z H W ) )
664adantr 451 . . . . . 6  |-  ( (
ph  /\  g  e.  ( Z H W ) )  ->  C  e.  Cat )
676adantr 451 . . . . . 6  |-  ( (
ph  /\  g  e.  ( Z H W ) )  ->  S  e.  B )
688adantr 451 . . . . . 6  |-  ( (
ph  /\  g  e.  ( Z H W ) )  ->  Z  e.  B )
6916adantr 451 . . . . . 6  |-  ( (
ph  /\  g  e.  ( Z H W ) )  ->  T  e.  B )
7012adantr 451 . . . . . 6  |-  ( (
ph  /\  g  e.  ( Z H W ) )  ->  P  e.  ( S H Z ) )
7121adantr 451 . . . . . . 7  |-  ( (
ph  /\  g  e.  ( Z H W ) )  ->  W  e.  B )
72 simpr 447 . . . . . . 7  |-  ( (
ph  /\  g  e.  ( Z H W ) )  ->  g  e.  ( Z H W ) )
7323adantr 451 . . . . . . 7  |-  ( (
ph  /\  g  e.  ( Z H W ) )  ->  Q  e.  ( W H T ) )
741, 2, 3, 66, 68, 71, 69, 72, 73catcocl 13603 . . . . . 6  |-  ( (
ph  /\  g  e.  ( Z H W ) )  ->  ( Q
( <. Z ,  W >. (comp `  C ) T ) g )  e.  ( Z H T ) )
751, 2, 3, 66, 67, 68, 69, 70, 74catcocl 13603 . . . . 5  |-  ( (
ph  /\  g  e.  ( Z H W ) )  ->  ( ( Q ( <. Z ,  W >. (comp `  C
) T ) g ) ( <. S ,  Z >. (comp `  C
) T ) P )  e.  ( S H T ) )
76 eqid 2296 . . . . 5  |-  ( g  e.  ( Z H W )  |->  ( ( Q ( <. Z ,  W >. (comp `  C
) T ) g ) ( <. S ,  Z >. (comp `  C
) T ) P ) )  =  ( g  e.  ( Z H W )  |->  ( ( Q ( <. Z ,  W >. (comp `  C ) T ) g ) ( <. S ,  Z >. (comp `  C ) T ) P ) )
7775, 76fmptd 5700 . . . 4  |-  ( ph  ->  ( g  e.  ( Z H W ) 
|->  ( ( Q (
<. Z ,  W >. (comp `  C ) T ) g ) ( <. S ,  Z >. (comp `  C ) T ) P ) ) : ( Z H W ) --> ( S H T ) )
7845, 46, 47, 56, 59, 62, 65, 77setcco 13931 . . 3  |-  ( ph  ->  ( ( g  e.  ( Z H W )  |->  ( ( Q ( <. Z ,  W >. (comp `  C ) T ) g ) ( <. S ,  Z >. (comp `  C ) T ) P ) ) ( <. ( X H Y ) ,  ( Z H W ) >. (comp `  D
) ( S H T ) ) ( f  e.  ( X H Y )  |->  ( ( L ( <. X ,  Y >. (comp `  C ) W ) f ) ( <. Z ,  X >. (comp `  C ) W ) K ) ) )  =  ( ( g  e.  ( Z H W )  |->  ( ( Q ( <. Z ,  W >. (comp `  C
) T ) g ) ( <. S ,  Z >. (comp `  C
) T ) P ) )  o.  (
f  e.  ( X H Y )  |->  ( ( L ( <. X ,  Y >. (comp `  C ) W ) f ) ( <. Z ,  X >. (comp `  C ) W ) K ) ) ) )
79 eqidd 2297 . . . 4  |-  ( ph  ->  ( f  e.  ( X H Y ) 
|->  ( ( L (
<. X ,  Y >. (comp `  C ) W ) f ) ( <. Z ,  X >. (comp `  C ) W ) K ) )  =  ( f  e.  ( X H Y ) 
|->  ( ( L (
<. X ,  Y >. (comp `  C ) W ) f ) ( <. Z ,  X >. (comp `  C ) W ) K ) ) )
80 eqidd 2297 . . . 4  |-  ( ph  ->  ( g  e.  ( Z H W ) 
|->  ( ( Q (
<. Z ,  W >. (comp `  C ) T ) g ) ( <. S ,  Z >. (comp `  C ) T ) P ) )  =  ( g  e.  ( Z H W ) 
|->  ( ( Q (
<. Z ,  W >. (comp `  C ) T ) g ) ( <. S ,  Z >. (comp `  C ) T ) P ) ) )
81 oveq2 5882 . . . . 5  |-  ( g  =  ( ( L ( <. X ,  Y >. (comp `  C ) W ) f ) ( <. Z ,  X >. (comp `  C ) W ) K )  ->  ( Q (
<. Z ,  W >. (comp `  C ) T ) g )  =  ( Q ( <. Z ,  W >. (comp `  C
) T ) ( ( L ( <. X ,  Y >. (comp `  C ) W ) f ) ( <. Z ,  X >. (comp `  C ) W ) K ) ) )
8281oveq1d 5889 . . . 4  |-  ( g  =  ( ( L ( <. X ,  Y >. (comp `  C ) W ) f ) ( <. Z ,  X >. (comp `  C ) W ) K )  ->  ( ( Q ( <. Z ,  W >. (comp `  C ) T ) g ) ( <. S ,  Z >. (comp `  C ) T ) P )  =  ( ( Q ( <. Z ,  W >. (comp `  C ) T ) ( ( L ( <. X ,  Y >. (comp `  C
) W ) f ) ( <. Z ,  X >. (comp `  C
) W ) K ) ) ( <. S ,  Z >. (comp `  C ) T ) P ) )
8363, 79, 80, 82fmptco 5707 . . 3  |-  ( ph  ->  ( ( g  e.  ( Z H W )  |->  ( ( Q ( <. Z ,  W >. (comp `  C ) T ) g ) ( <. S ,  Z >. (comp `  C ) T ) P ) )  o.  ( f  e.  ( X H Y )  |->  ( ( L ( <. X ,  Y >. (comp `  C
) W ) f ) ( <. Z ,  X >. (comp `  C
) W ) K ) ) )  =  ( f  e.  ( X H Y ) 
|->  ( ( Q (
<. Z ,  W >. (comp `  C ) T ) ( ( L (
<. X ,  Y >. (comp `  C ) W ) f ) ( <. Z ,  X >. (comp `  C ) W ) K ) ) (
<. S ,  Z >. (comp `  C ) T ) P ) ) )
8444, 78, 833eqtrd 2332 . 2  |-  ( ph  ->  ( ( P (
<. Z ,  W >. ( 2nd `  M )
<. S ,  T >. ) Q ) ( <.
( X H Y ) ,  ( Z H W ) >.
(comp `  D )
( S H T ) ) ( K ( <. X ,  Y >. ( 2nd `  M
) <. Z ,  W >. ) L ) )  =  ( f  e.  ( X H Y )  |->  ( ( Q ( <. Z ,  W >. (comp `  C ) T ) ( ( L ( <. X ,  Y >. (comp `  C
) W ) f ) ( <. Z ,  X >. (comp `  C
) W ) K ) ) ( <. S ,  Z >. (comp `  C ) T ) P ) ) )
8538, 41, 843eqtr4d 2338 1  |-  ( ph  ->  ( ( K (
<. S ,  Z >. (comp `  C ) X ) P ) ( <. X ,  Y >. ( 2nd `  M )
<. S ,  T >. ) ( Q ( <. Y ,  W >. (comp `  C ) T ) L ) )  =  ( ( P (
<. Z ,  W >. ( 2nd `  M )
<. S ,  T >. ) Q ) ( <.
( X H Y ) ,  ( Z H W ) >.
(comp `  D )
( S H T ) ) ( K ( <. X ,  Y >. ( 2nd `  M
) <. Z ,  W >. ) L ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   <.cop 3656    e. cmpt 4093    X. cxp 4703   ran crn 4706    o. ccom 4709    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   2ndc2nd 6137   Basecbs 13164    Hom chom 13235  compcco 13236   Catccat 13582    Homf chomf 13584  oppCatcoppc 13630   SetCatcsetc 13923  HomFchof 14038
This theorem is referenced by:  hofcl  14049
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  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2462  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-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  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-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-hom 13248  df-cco 13249  df-cat 13586  df-homf 13588  df-setc 13924  df-hof 14040
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