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Theorem hofcllem 15402
Description: Lemma for hofcl 15403. (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 2467 . . . . 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 14957 . . . . . . 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 14957 . . . . 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 14958 . . . 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 14958 . . . . . . 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 6310 . . . . . 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 14957 . . . . . . 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 14958 . . . . . 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 2508 . . . . 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 6310 . . . 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 2510 . . 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 4539 . 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 14957 . . 3  |-  ( ph  ->  ( K ( <. S ,  Z >. (comp `  C ) X ) P )  e.  ( S H X ) )
4139, 4, 1, 2, 10, 18, 6, 16, 3, 40, 24hof2val 15400 . 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 15400 . . . 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 15400 . . . 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 6313 . . 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 2467 . . . 4  |-  (comp `  D )  =  (comp `  D )
48 eqid 2467 . . . . . 6  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
4948, 1, 2, 10, 18homfval 14965 . . . . 5  |-  ( ph  ->  ( X ( Hom f  `  C ) Y )  =  ( X H Y ) )
5048, 1homffn 14966 . . . . . . . 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 5598 . . . . . . 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 6442 . . . . 5  |-  ( ph  ->  ( X ( Hom f  `  C ) Y )  e.  U )
5649, 55eqeltrrd 2556 . . . 4  |-  ( ph  ->  ( X H Y )  e.  U )
5748, 1, 2, 8, 21homfval 14965 . . . . 5  |-  ( ph  ->  ( Z ( Hom f  `  C ) W )  =  ( Z H W ) )
5854, 8, 21fovrnd 6442 . . . . 5  |-  ( ph  ->  ( Z ( Hom f  `  C ) W )  e.  U )
5957, 58eqeltrrd 2556 . . . 4  |-  ( ph  ->  ( Z H W )  e.  U )
6048, 1, 2, 6, 16homfval 14965 . . . . 5  |-  ( ph  ->  ( S ( Hom f  `  C ) T )  =  ( S H T ) )
6154, 6, 16fovrnd 6442 . . . . 5  |-  ( ph  ->  ( S ( Hom f  `  C ) T )  e.  U )
6260, 61eqeltrrd 2556 . . . 4  |-  ( ph  ->  ( S H T )  e.  U )
631, 2, 3, 5, 9, 11, 28, 15, 33catcocl 14957 . . . . 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 2467 . . . . 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 6056 . . . 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 14957 . . . . . 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 14957 . . . . 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 2467 . . . . 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 6056 . . . 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 15285 . . 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 2468 . . . 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 2468 . . . 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 6303 . . . . 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 6310 . . . 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 6065 . . 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 2512 . 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 2518 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 1379    e. wcel 1767    C_ wss 3481   <.cop 4039    |-> cmpt 4511    X. cxp 5003   ran crn 5006    o. ccom 5009    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295   2ndc2nd 6794   Basecbs 14507   Hom chom 14583  compcco 14584   Catccat 14936   Hom f chomf 14938  oppCatcoppc 14984   SetCatcsetc 15277  HomFchof 15392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-hom 14596  df-cco 14597  df-cat 14940  df-homf 14942  df-setc 15278  df-hof 15394
This theorem is referenced by:  hofcl  15403
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