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Theorem fuccocl 14163
Description: The composition of two natural transformations is a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
fuccocl.q  |-  Q  =  ( C FuncCat  D )
fuccocl.n  |-  N  =  ( C Nat  D )
fuccocl.x  |-  .xb  =  (comp `  Q )
fuccocl.r  |-  ( ph  ->  R  e.  ( F N G ) )
fuccocl.s  |-  ( ph  ->  S  e.  ( G N H ) )
Assertion
Ref Expression
fuccocl  |-  ( ph  ->  ( S ( <. F ,  G >.  .xb 
H ) R )  e.  ( F N H ) )

Proof of Theorem fuccocl
Dummy variables  x  f  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fuccocl.q . . . 4  |-  Q  =  ( C FuncCat  D )
2 fuccocl.n . . . 4  |-  N  =  ( C Nat  D )
3 eqid 2438 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
4 eqid 2438 . . . 4  |-  (comp `  D )  =  (comp `  D )
5 fuccocl.x . . . 4  |-  .xb  =  (comp `  Q )
6 fuccocl.r . . . 4  |-  ( ph  ->  R  e.  ( F N G ) )
7 fuccocl.s . . . 4  |-  ( ph  ->  S  e.  ( G N H ) )
81, 2, 3, 4, 5, 6, 7fucco 14161 . . 3  |-  ( ph  ->  ( S ( <. F ,  G >.  .xb 
H ) R )  =  ( x  e.  ( Base `  C
)  |->  ( ( S `
 x ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  x
) ) ( R `
 x ) ) ) )
9 eqid 2438 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
10 eqid 2438 . . . . . 6  |-  (  Hom  `  D )  =  (  Hom  `  D )
112natrcl 14149 . . . . . . . . . . 11  |-  ( R  e.  ( F N G )  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D
) ) )
126, 11syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D ) ) )
1312simpld 447 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
14 funcrcl 14062 . . . . . . . . 9  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
1513, 14syl 16 . . . . . . . 8  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
1615simprd 451 . . . . . . 7  |-  ( ph  ->  D  e.  Cat )
1716adantr 453 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  D  e.  Cat )
18 relfunc 14061 . . . . . . . . 9  |-  Rel  ( C  Func  D )
19 1st2ndbr 6398 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
2018, 13, 19sylancr 646 . . . . . . . 8  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
213, 9, 20funcf1 14065 . . . . . . 7  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
2221ffvelrnda 5872 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
232natrcl 14149 . . . . . . . . . . 11  |-  ( S  e.  ( G N H )  ->  ( G  e.  ( C  Func  D )  /\  H  e.  ( C  Func  D
) ) )
247, 23syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( G  e.  ( C  Func  D )  /\  H  e.  ( C  Func  D ) ) )
2524simpld 447 . . . . . . . . 9  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
26 1st2ndbr 6398 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 1st `  G ) ( C  Func  D )
( 2nd `  G
) )
2718, 25, 26sylancr 646 . . . . . . . 8  |-  ( ph  ->  ( 1st `  G
) ( C  Func  D ) ( 2nd `  G
) )
283, 9, 27funcf1 14065 . . . . . . 7  |-  ( ph  ->  ( 1st `  G
) : ( Base `  C ) --> ( Base `  D ) )
2928ffvelrnda 5872 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  G ) `  x )  e.  (
Base `  D )
)
3024simprd 451 . . . . . . . . 9  |-  ( ph  ->  H  e.  ( C 
Func  D ) )
31 1st2ndbr 6398 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  D )  /\  H  e.  ( C  Func  D
) )  ->  ( 1st `  H ) ( C  Func  D )
( 2nd `  H
) )
3218, 30, 31sylancr 646 . . . . . . . 8  |-  ( ph  ->  ( 1st `  H
) ( C  Func  D ) ( 2nd `  H
) )
333, 9, 32funcf1 14065 . . . . . . 7  |-  ( ph  ->  ( 1st `  H
) : ( Base `  C ) --> ( Base `  D ) )
3433ffvelrnda 5872 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  H ) `  x )  e.  (
Base `  D )
)
352, 6nat1st2nd 14150 . . . . . . . 8  |-  ( ph  ->  R  e.  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
3635adantr 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  R  e.  ( <. ( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
37 simpr 449 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  x  e.  ( Base `  C )
)
382, 36, 3, 10, 37natcl 14152 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( R `  x )  e.  ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  G
) `  x )
) )
392, 7nat1st2nd 14150 . . . . . . . 8  |-  ( ph  ->  S  e.  ( <.
( 1st `  G
) ,  ( 2nd `  G ) >. N <. ( 1st `  H ) ,  ( 2nd `  H
) >. ) )
4039adantr 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  S  e.  ( <. ( 1st `  G
) ,  ( 2nd `  G ) >. N <. ( 1st `  H ) ,  ( 2nd `  H
) >. ) )
412, 40, 3, 10, 37natcl 14152 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( S `  x )  e.  ( ( ( 1st `  G
) `  x )
(  Hom  `  D ) ( ( 1st `  H
) `  x )
) )
429, 10, 4, 17, 22, 29, 34, 38, 41catcocl 13912 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( S `  x )
( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  H
) `  x )
) ( R `  x ) )  e.  ( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  H ) `  x
) ) )
4342ralrimiva 2791 . . . 4  |-  ( ph  ->  A. x  e.  (
Base `  C )
( ( S `  x ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  x
) ) ( R `
 x ) )  e.  ( ( ( 1st `  F ) `
 x ) (  Hom  `  D )
( ( 1st `  H
) `  x )
) )
44 fvex 5744 . . . . 5  |-  ( Base `  C )  e.  _V
45 mptelixpg 7101 . . . . 5  |-  ( (
Base `  C )  e.  _V  ->  ( (
x  e.  ( Base `  C )  |->  ( ( S `  x ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  H
) `  x )
) ( R `  x ) ) )  e.  X_ x  e.  (
Base `  C )
( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  H ) `  x
) )  <->  A. x  e.  ( Base `  C
) ( ( S `
 x ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  x
) ) ( R `
 x ) )  e.  ( ( ( 1st `  F ) `
 x ) (  Hom  `  D )
( ( 1st `  H
) `  x )
) ) )
4644, 45ax-mp 8 . . . 4  |-  ( ( x  e.  ( Base `  C )  |->  ( ( S `  x ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  H
) `  x )
) ( R `  x ) ) )  e.  X_ x  e.  (
Base `  C )
( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  H ) `  x
) )  <->  A. x  e.  ( Base `  C
) ( ( S `
 x ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  x
) ) ( R `
 x ) )  e.  ( ( ( 1st `  F ) `
 x ) (  Hom  `  D )
( ( 1st `  H
) `  x )
) )
4743, 46sylibr 205 . . 3  |-  ( ph  ->  ( x  e.  (
Base `  C )  |->  ( ( S `  x ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  x
) ) ( R `
 x ) ) )  e.  X_ x  e.  ( Base `  C
) ( ( ( 1st `  F ) `
 x ) (  Hom  `  D )
( ( 1st `  H
) `  x )
) )
488, 47eqeltrd 2512 . 2  |-  ( ph  ->  ( S ( <. F ,  G >.  .xb 
H ) R )  e.  X_ x  e.  (
Base `  C )
( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  H ) `  x
) ) )
4916adantr 453 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  D  e.  Cat )
5021adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( 1st `  F ) : (
Base `  C ) --> ( Base `  D )
)
51 simpr1 964 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  x  e.  ( Base `  C )
)
5250, 51ffvelrnd 5873 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
53 simpr2 965 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  y  e.  ( Base `  C )
)
5450, 53ffvelrnd 5873 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( ( 1st `  F ) `  y )  e.  (
Base `  D )
)
5528adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( 1st `  G ) : (
Base `  C ) --> ( Base `  D )
)
5655, 53ffvelrnd 5873 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( ( 1st `  G ) `  y )  e.  (
Base `  D )
)
57 eqid 2438 . . . . . . . 8  |-  (  Hom  `  C )  =  (  Hom  `  C )
5820adantr 453 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( 1st `  F ) ( C 
Func  D ) ( 2nd `  F ) )
593, 57, 10, 58, 51, 53funcf2 14067 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( x
( 2nd `  F
) y ) : ( x (  Hom  `  C ) y ) --> ( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  F ) `  y
) ) )
60 simpr3 966 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  f  e.  ( x (  Hom  `  C ) y ) )
6159, 60ffvelrnd 5873 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( (
x ( 2nd `  F
) y ) `  f )  e.  ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
) )
6235adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  R  e.  ( <. ( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
632, 62, 3, 10, 53natcl 14152 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( R `  y )  e.  ( ( ( 1st `  F
) `  y )
(  Hom  `  D ) ( ( 1st `  G
) `  y )
) )
6433adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( 1st `  H ) : (
Base `  C ) --> ( Base `  D )
)
6564, 53ffvelrnd 5873 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( ( 1st `  H ) `  y )  e.  (
Base `  D )
)
6639adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  S  e.  ( <. ( 1st `  G
) ,  ( 2nd `  G ) >. N <. ( 1st `  H ) ,  ( 2nd `  H
) >. ) )
672, 66, 3, 10, 53natcl 14152 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( S `  y )  e.  ( ( ( 1st `  G
) `  y )
(  Hom  `  D ) ( ( 1st `  H
) `  y )
) )
689, 10, 4, 49, 52, 54, 56, 61, 63, 65, 67catass 13913 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( (
( S `  y
) ( <. (
( 1st `  F
) `  y ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( R `
 y ) ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  F ) `
 y ) >.
(comp `  D )
( ( 1st `  H
) `  y )
) ( ( x ( 2nd `  F
) y ) `  f ) )  =  ( ( S `  y ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( R `  y ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  F ) `
 y ) >.
(comp `  D )
( ( 1st `  G
) `  y )
) ( ( x ( 2nd `  F
) y ) `  f ) ) ) )
692, 62, 3, 57, 4, 51, 53, 60nati 14154 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( ( R `  y )
( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  F ) `
 y ) >.
(comp `  D )
( ( 1st `  G
) `  y )
) ( ( x ( 2nd `  F
) y ) `  f ) )  =  ( ( ( x ( 2nd `  G
) y ) `  f ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  G ) `  y
) ) ( R `
 x ) ) )
7069oveq2d 6099 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( ( S `  y )
( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 y ) >.
(comp `  D )
( ( 1st `  H
) `  y )
) ( ( R `
 y ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  G ) `  y
) ) ( ( x ( 2nd `  F
) y ) `  f ) ) )  =  ( ( S `
 y ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( ( x ( 2nd `  G ) y ) `
 f ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  G ) `  y
) ) ( R `
 x ) ) ) )
7155, 51ffvelrnd 5873 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( ( 1st `  G ) `  x )  e.  (
Base `  D )
)
722, 62, 3, 10, 51natcl 14152 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( R `  x )  e.  ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  G
) `  x )
) )
7327adantr 453 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( 1st `  G ) ( C 
Func  D ) ( 2nd `  G ) )
743, 57, 10, 73, 51, 53funcf2 14067 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( x
( 2nd `  G
) y ) : ( x (  Hom  `  C ) y ) --> ( ( ( 1st `  G ) `  x
) (  Hom  `  D
) ( ( 1st `  G ) `  y
) ) )
7574, 60ffvelrnd 5873 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( (
x ( 2nd `  G
) y ) `  f )  e.  ( ( ( 1st `  G
) `  x )
(  Hom  `  D ) ( ( 1st `  G
) `  y )
) )
769, 10, 4, 49, 52, 71, 56, 72, 75, 65, 67catass 13913 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( (
( S `  y
) ( <. (
( 1st `  G
) `  x ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( x ( 2nd `  G
) y ) `  f ) ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( R `
 x ) )  =  ( ( S `
 y ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( ( x ( 2nd `  G ) y ) `
 f ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  G ) `  y
) ) ( R `
 x ) ) ) )
772, 66, 3, 57, 4, 51, 53, 60nati 14154 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( ( S `  y )
( <. ( ( 1st `  G ) `  x
) ,  ( ( 1st `  G ) `
 y ) >.
(comp `  D )
( ( 1st `  H
) `  y )
) ( ( x ( 2nd `  G
) y ) `  f ) )  =  ( ( ( x ( 2nd `  H
) y ) `  f ) ( <.
( ( 1st `  G
) `  x ) ,  ( ( 1st `  H ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( S `
 x ) ) )
7877oveq1d 6098 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( (
( S `  y
) ( <. (
( 1st `  G
) `  x ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( x ( 2nd `  G
) y ) `  f ) ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( R `
 x ) )  =  ( ( ( ( x ( 2nd `  H ) y ) `
 f ) (
<. ( ( 1st `  G
) `  x ) ,  ( ( 1st `  H ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( S `
 x ) ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  H
) `  y )
) ( R `  x ) ) )
7970, 76, 783eqtr2d 2476 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( ( S `  y )
( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 y ) >.
(comp `  D )
( ( 1st `  H
) `  y )
) ( ( R `
 y ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  G ) `  y
) ) ( ( x ( 2nd `  F
) y ) `  f ) ) )  =  ( ( ( ( x ( 2nd `  H ) y ) `
 f ) (
<. ( ( 1st `  G
) `  x ) ,  ( ( 1st `  H ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( S `
 x ) ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  H
) `  y )
) ( R `  x ) ) )
8064, 51ffvelrnd 5873 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( ( 1st `  H ) `  x )  e.  (
Base `  D )
)
812, 66, 3, 10, 51natcl 14152 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( S `  x )  e.  ( ( ( 1st `  G
) `  x )
(  Hom  `  D ) ( ( 1st `  H
) `  x )
) )
8232adantr 453 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( 1st `  H ) ( C 
Func  D ) ( 2nd `  H ) )
833, 57, 10, 82, 51, 53funcf2 14067 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( x
( 2nd `  H
) y ) : ( x (  Hom  `  C ) y ) --> ( ( ( 1st `  H ) `  x
) (  Hom  `  D
) ( ( 1st `  H ) `  y
) ) )
8483, 60ffvelrnd 5873 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( (
x ( 2nd `  H
) y ) `  f )  e.  ( ( ( 1st `  H
) `  x )
(  Hom  `  D ) ( ( 1st `  H
) `  y )
) )
859, 10, 4, 49, 52, 71, 80, 72, 81, 65, 84catass 13913 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( (
( ( x ( 2nd `  H ) y ) `  f
) ( <. (
( 1st `  G
) `  x ) ,  ( ( 1st `  H ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( S `
 x ) ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  H
) `  y )
) ( R `  x ) )  =  ( ( ( x ( 2nd `  H
) y ) `  f ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  H ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( S `  x ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  H
) `  x )
) ( R `  x ) ) ) )
8668, 79, 853eqtrd 2474 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( (
( S `  y
) ( <. (
( 1st `  F
) `  y ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( R `
 y ) ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  F ) `
 y ) >.
(comp `  D )
( ( 1st `  H
) `  y )
) ( ( x ( 2nd `  F
) y ) `  f ) )  =  ( ( ( x ( 2nd `  H
) y ) `  f ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  H ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( S `  x ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  H
) `  x )
) ( R `  x ) ) ) )
876adantr 453 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  R  e.  ( F N G ) )
887adantr 453 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  S  e.  ( G N H ) )
891, 2, 3, 4, 5, 87, 88, 53fuccoval 14162 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( ( S ( <. F ,  G >.  .xb  H ) R ) `  y )  =  ( ( S `
 y ) (
<. ( ( 1st `  F
) `  y ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( R `
 y ) ) )
9089oveq1d 6098 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( (
( S ( <. F ,  G >.  .xb 
H ) R ) `
 y ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( x ( 2nd `  F
) y ) `  f ) )  =  ( ( ( S `
 y ) (
<. ( ( 1st `  F
) `  y ) ,  ( ( 1st `  G ) `  y
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( R `
 y ) ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  F ) `
 y ) >.
(comp `  D )
( ( 1st `  H
) `  y )
) ( ( x ( 2nd `  F
) y ) `  f ) ) )
911, 2, 3, 4, 5, 87, 88, 51fuccoval 14162 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( ( S ( <. F ,  G >.  .xb  H ) R ) `  x )  =  ( ( S `
 x ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  G ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  x
) ) ( R `
 x ) ) )
9291oveq2d 6099 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( (
( x ( 2nd `  H ) y ) `
 f ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  H ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( S ( <. F ,  G >.  .xb  H ) R ) `  x ) )  =  ( ( ( x ( 2nd `  H ) y ) `
 f ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  H ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( S `  x ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  G ) `
 x ) >.
(comp `  D )
( ( 1st `  H
) `  x )
) ( R `  x ) ) ) )
9386, 90, 923eqtr4d 2480 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( (
( S ( <. F ,  G >.  .xb 
H ) R ) `
 y ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( x ( 2nd `  F
) y ) `  f ) )  =  ( ( ( x ( 2nd `  H
) y ) `  f ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  H ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( S ( <. F ,  G >.  .xb  H ) R ) `  x ) ) )
9493ralrimivvva 2801 . 2  |-  ( ph  ->  A. x  e.  (
Base `  C ) A. y  e.  ( Base `  C ) A. f  e.  ( x
(  Hom  `  C ) y ) ( ( ( S ( <. F ,  G >.  .xb 
H ) R ) `
 y ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( x ( 2nd `  F
) y ) `  f ) )  =  ( ( ( x ( 2nd `  H
) y ) `  f ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  H ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( S ( <. F ,  G >.  .xb  H ) R ) `  x ) ) )
952, 3, 57, 10, 4, 13, 30isnat2 14147 . 2  |-  ( ph  ->  ( ( S (
<. F ,  G >.  .xb 
H ) R )  e.  ( F N H )  <->  ( ( S ( <. F ,  G >.  .xb  H ) R )  e.  X_ x  e.  ( Base `  C
) ( ( ( 1st `  F ) `
 x ) (  Hom  `  D )
( ( 1st `  H
) `  x )
)  /\  A. x  e.  ( Base `  C
) A. y  e.  ( Base `  C
) A. f  e.  ( x (  Hom  `  C ) y ) ( ( ( S ( <. F ,  G >. 
.xb  H ) R ) `  y ) ( <. ( ( 1st `  F ) `  x
) ,  ( ( 1st `  F ) `
 y ) >.
(comp `  D )
( ( 1st `  H
) `  y )
) ( ( x ( 2nd `  F
) y ) `  f ) )  =  ( ( ( x ( 2nd `  H
) y ) `  f ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  H ) `  x
) >. (comp `  D
) ( ( 1st `  H ) `  y
) ) ( ( S ( <. F ,  G >.  .xb  H ) R ) `  x ) ) ) ) )
9648, 94, 95mpbir2and 890 1  |-  ( ph  ->  ( S ( <. F ,  G >.  .xb 
H ) R )  e.  ( F N H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958   <.cop 3819   class class class wbr 4214    e. cmpt 4268   Rel wrel 4885   -->wf 5452   ` cfv 5456  (class class class)co 6083   1stc1st 6349   2ndc2nd 6350   X_cixp 7065   Basecbs 13471    Hom chom 13542  compcco 13543   Catccat 13891    Func cfunc 14053   Nat cnat 14140   FuncCat cfuc 14141
This theorem is referenced by:  fucass  14167  fuccatid  14168  evlfcllem  14320  yonedalem3b  14378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-map 7022  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-fz 11046  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-hom 13555  df-cco 13556  df-cat 13895  df-func 14057  df-nat 14142  df-fuc 14143
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