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Theorem hofcllem 15190
Description: Lemma for hofcl 15191. (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  ( Hom f  `  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 2454 . . . . 5  |-  (comp `  C )  =  (comp `  C )
4 hofcl.c . . . . . 6  |-  ( ph  ->  C  e.  Cat )
54adantr 465 . . . . 5  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  C  e.  Cat )
6 hofcllem.s . . . . . 6  |-  ( ph  ->  S  e.  B )
76adantr 465 . . . . 5  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  S  e.  B )
8 hofcllem.z . . . . . 6  |-  ( ph  ->  Z  e.  B )
98adantr 465 . . . . 5  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  Z  e.  B )
10 hofcllem.x . . . . . 6  |-  ( ph  ->  X  e.  B )
1110adantr 465 . . . . 5  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  X  e.  B )
12 hofcllem.p . . . . . 6  |-  ( ph  ->  P  e.  ( S H Z ) )
1312adantr 465 . . . . 5  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  P  e.  ( S H Z ) )
14 hofcllem.m . . . . . 6  |-  ( ph  ->  K  e.  ( Z H X ) )
1514adantr 465 . . . . 5  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  K  e.  ( Z H X ) )
16 hofcllem.t . . . . . 6  |-  ( ph  ->  T  e.  B )
1716adantr 465 . . . . 5  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  T  e.  B )
18 hofcllem.y . . . . . . 7  |-  ( ph  ->  Y  e.  B )
1918adantr 465 . . . . . 6  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  Y  e.  B )
20 simpr 461 . . . . . 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 14745 . . . . . . 7  |-  ( ph  ->  ( Q ( <. Y ,  W >. (comp `  C ) T ) L )  e.  ( Y H T ) )
2524adantr 465 . . . . . 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 14745 . . . . 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 14746 . . . 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 465 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  W  e.  B )
2922adantr 465 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  L  e.  ( Y H W ) )
3023adantr 465 . . . . . . . 8  |-  ( (
ph  /\  f  e.  ( X H Y ) )  ->  Q  e.  ( W H T ) )
311, 2, 3, 5, 11, 19, 28, 20, 29, 17, 30catass 14746 . . . . . . 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 6218 . . . . . 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 14745 . . . . . . 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 14746 . . . . . 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 2495 . . . . 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 6218 . . . 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 2497 . . 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 4489 . 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 14745 . . 3  |-  ( ph  ->  ( K ( <. S ,  Z >. (comp `  C ) X ) P )  e.  ( S H X ) )
4139, 4, 1, 2, 10, 18, 6, 16, 3, 40, 24hof2val 15188 . 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 15188 . . . 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 15188 . . . 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 6221 . . 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 2454 . . . 4  |-  (comp `  D )  =  (comp `  D )
48 eqid 2454 . . . . . 6  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
4948, 1, 2, 10, 18homfval 14753 . . . . 5  |-  ( ph  ->  ( X ( Hom f  `  C ) Y )  =  ( X H Y ) )
5048, 1homffn 14754 . . . . . . . 8  |-  ( Hom f  `  C )  Fn  ( B  X.  B )
5150a1i 11 . . . . . . 7  |-  ( ph  ->  ( Hom f  `  C )  Fn  ( B  X.  B
) )
52 hofcl.h . . . . . . 7  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  U )
53 df-f 5533 . . . . . . 7  |-  ( ( Hom f  `  C ) : ( B  X.  B ) --> U  <->  ( ( Hom f  `  C )  Fn  ( B  X.  B )  /\  ran  ( Hom f  `  C )  C_  U ) )
5451, 52, 53sylanbrc 664 . . . . . 6  |-  ( ph  ->  ( Hom f  `  C ) : ( B  X.  B
) --> U )
5554, 10, 18fovrnd 6348 . . . . 5  |-  ( ph  ->  ( X ( Hom f  `  C ) Y )  e.  U )
5649, 55eqeltrrd 2543 . . . 4  |-  ( ph  ->  ( X H Y )  e.  U )
5748, 1, 2, 8, 21homfval 14753 . . . . 5  |-  ( ph  ->  ( Z ( Hom f  `  C ) W )  =  ( Z H W ) )
5854, 8, 21fovrnd 6348 . . . . 5  |-  ( ph  ->  ( Z ( Hom f  `  C ) W )  e.  U )
5957, 58eqeltrrd 2543 . . . 4  |-  ( ph  ->  ( Z H W )  e.  U )
6048, 1, 2, 6, 16homfval 14753 . . . . 5  |-  ( ph  ->  ( S ( Hom f  `  C ) T )  =  ( S H T ) )
6154, 6, 16fovrnd 6348 . . . . 5  |-  ( ph  ->  ( S ( Hom f  `  C ) T )  e.  U )
6260, 61eqeltrrd 2543 . . . 4  |-  ( ph  ->  ( S H T )  e.  U )
631, 2, 3, 5, 9, 11, 28, 15, 33catcocl 14745 . . . . 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 2454 . . . . 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 5979 . . . 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 465 . . . . . 6  |-  ( (
ph  /\  g  e.  ( Z H W ) )  ->  C  e.  Cat )
676adantr 465 . . . . . 6  |-  ( (
ph  /\  g  e.  ( Z H W ) )  ->  S  e.  B )
688adantr 465 . . . . . 6  |-  ( (
ph  /\  g  e.  ( Z H W ) )  ->  Z  e.  B )
6916adantr 465 . . . . . 6  |-  ( (
ph  /\  g  e.  ( Z H W ) )  ->  T  e.  B )
7012adantr 465 . . . . . 6  |-  ( (
ph  /\  g  e.  ( Z H W ) )  ->  P  e.  ( S H Z ) )
7121adantr 465 . . . . . . 7  |-  ( (
ph  /\  g  e.  ( Z H W ) )  ->  W  e.  B )
72 simpr 461 . . . . . . 7  |-  ( (
ph  /\  g  e.  ( Z H W ) )  ->  g  e.  ( Z H W ) )
7323adantr 465 . . . . . . 7  |-  ( (
ph  /\  g  e.  ( Z H W ) )  ->  Q  e.  ( W H T ) )
741, 2, 3, 66, 68, 71, 69, 72, 73catcocl 14745 . . . . . 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 14745 . . . . 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 2454 . . . . 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 5979 . . . 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 15073 . . 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 2455 . . . 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 2455 . . . 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 6211 . . . . 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 6218 . . . 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 5988 . . 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 2499 . 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 2505 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    C_ wss 3439   <.cop 3994    |-> cmpt 4461    X. cxp 4949   ran crn 4952    o. ccom 4955    Fn wfn 5524   -->wf 5525   ` cfv 5529  (class class class)co 6203   2ndc2nd 6689   Basecbs 14295   Hom chom 14371  compcco 14372   Catccat 14724   Hom f chomf 14726  oppCatcoppc 14772   SetCatcsetc 15065  HomFchof 15180
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-3 10495  df-4 10496  df-5 10497  df-6 10498  df-7 10499  df-8 10500  df-9 10501  df-10 10502  df-n0 10694  df-z 10761  df-dec 10870  df-uz 10976  df-fz 11558  df-struct 14297  df-ndx 14298  df-slot 14299  df-base 14300  df-hom 14384  df-cco 14385  df-cat 14728  df-homf 14730  df-setc 15066  df-hof 15182
This theorem is referenced by:  hofcl  15191
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