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Theorem fuccocl 13838
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 2283 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
4 eqid 2283 . . . 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 13836 . . 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 2283 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
10 eqid 2283 . . . . . 6  |-  (  Hom  `  D )  =  (  Hom  `  D )
112natrcl 13824 . . . . . . . . . . 11  |-  ( R  e.  ( F N G )  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D
) ) )
126, 11syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( F  e.  ( C  Func  D )  /\  G  e.  ( C  Func  D ) ) )
1312simpld 445 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
14 funcrcl 13737 . . . . . . . . 9  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
1513, 14syl 15 . . . . . . . 8  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
1615simprd 449 . . . . . . 7  |-  ( ph  ->  D  e.  Cat )
1716adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  D  e.  Cat )
18 relfunc 13736 . . . . . . . . 9  |-  Rel  ( C  Func  D )
19 1st2ndbr 6169 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
2018, 13, 19sylancr 644 . . . . . . . 8  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
213, 9, 20funcf1 13740 . . . . . . 7  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
2221ffvelrnda 5665 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
232natrcl 13824 . . . . . . . . . . 11  |-  ( S  e.  ( G N H )  ->  ( G  e.  ( C  Func  D )  /\  H  e.  ( C  Func  D
) ) )
247, 23syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( G  e.  ( C  Func  D )  /\  H  e.  ( C  Func  D ) ) )
2524simpld 445 . . . . . . . . 9  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
26 1st2ndbr 6169 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 1st `  G ) ( C  Func  D )
( 2nd `  G
) )
2718, 25, 26sylancr 644 . . . . . . . 8  |-  ( ph  ->  ( 1st `  G
) ( C  Func  D ) ( 2nd `  G
) )
283, 9, 27funcf1 13740 . . . . . . 7  |-  ( ph  ->  ( 1st `  G
) : ( Base `  C ) --> ( Base `  D ) )
2928ffvelrnda 5665 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  G ) `  x )  e.  (
Base `  D )
)
3024simprd 449 . . . . . . . . 9  |-  ( ph  ->  H  e.  ( C 
Func  D ) )
31 1st2ndbr 6169 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  D )  /\  H  e.  ( C  Func  D
) )  ->  ( 1st `  H ) ( C  Func  D )
( 2nd `  H
) )
3218, 30, 31sylancr 644 . . . . . . . 8  |-  ( ph  ->  ( 1st `  H
) ( C  Func  D ) ( 2nd `  H
) )
333, 9, 32funcf1 13740 . . . . . . 7  |-  ( ph  ->  ( 1st `  H
) : ( Base `  C ) --> ( Base `  D ) )
3433ffvelrnda 5665 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  H ) `  x )  e.  (
Base `  D )
)
352, 6nat1st2nd 13825 . . . . . . . 8  |-  ( ph  ->  R  e.  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
3635adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  R  e.  ( <. ( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
37 simpr 447 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  x  e.  ( Base `  C )
)
382, 36, 3, 10, 37natcl 13827 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( R `  x )  e.  ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  G
) `  x )
) )
392, 7nat1st2nd 13825 . . . . . . . 8  |-  ( ph  ->  S  e.  ( <.
( 1st `  G
) ,  ( 2nd `  G ) >. N <. ( 1st `  H ) ,  ( 2nd `  H
) >. ) )
4039adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  S  e.  ( <. ( 1st `  G
) ,  ( 2nd `  G ) >. N <. ( 1st `  H ) ,  ( 2nd `  H
) >. ) )
412, 40, 3, 10, 37natcl 13827 . . . . . 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 13587 . . . . 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 2626 . . . 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 5539 . . . . 5  |-  ( Base `  C )  e.  _V
45 mptelixpg 6853 . . . . 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 203 . . 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 2357 . 2  |-  ( ph  ->  ( S ( <. F ,  G >.  .xb 
H ) R )  e.  X_ x  e.  (
Base `  C )
( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  H ) `  x
) ) )
4916adantr 451 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  D  e.  Cat )
5021adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( 1st `  F ) : (
Base `  C ) --> ( Base `  D )
)
51 simpr1 961 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  x  e.  ( Base `  C )
)
5250, 51ffvelrnd 5666 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
53 simpr2 962 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  y  e.  ( Base `  C )
)
5450, 53ffvelrnd 5666 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( ( 1st `  F ) `  y )  e.  (
Base `  D )
)
5528adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( 1st `  G ) : (
Base `  C ) --> ( Base `  D )
)
5655, 53ffvelrnd 5666 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( ( 1st `  G ) `  y )  e.  (
Base `  D )
)
57 eqid 2283 . . . . . . . 8  |-  (  Hom  `  C )  =  (  Hom  `  C )
5820adantr 451 . . . . . . . 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 13742 . . . . . . 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 963 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  f  e.  ( x (  Hom  `  C ) y ) )
6159, 60ffvelrnd 5666 . . . . . 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 451 . . . . . . 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 13827 . . . . . 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 451 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( 1st `  H ) : (
Base `  C ) --> ( Base `  D )
)
6564, 53ffvelrnd 5666 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  ( ( 1st `  H ) `  y )  e.  (
Base `  D )
)
6639adantr 451 . . . . . . 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 13827 . . . . . 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 13588 . . . . 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 13829 . . . . . . 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 5874 . . . . . 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 5666 . . . . . . 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 13827 . . . . . . 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 451 . . . . . . . . 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 13742 . . . . . . . 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 5666 . . . . . . 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 13588 . . . . . 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 13829 . . . . . . 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 5873 . . . . . 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 2321 . . . . 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 5666 . . . . . 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 13827 . . . . . 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 451 . . . . . . . 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 13742 . . . . . . 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 5666 . . . . . 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 13588 . . . . 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 2319 . . . 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 451 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x (  Hom  `  C
) y ) ) )  ->  R  e.  ( F N G ) )
887adantr 451 . . . . . 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 13837 . . . . 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 5873 . . . 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 13837 . . . . 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 5874 . . . 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 2325 . . 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 2636 . 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 13822 . 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 888 1  |-  ( ph  ->  ( S ( <. F ,  G >.  .xb 
H ) R )  e.  ( F N H ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   <.cop 3643   class class class wbr 4023    e. cmpt 4077   Rel wrel 4694   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   X_cixp 6817   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566    Func cfunc 13728   Nat cnat 13815   FuncCat cfuc 13816
This theorem is referenced by:  fucass  13842  fuccatid  13843  evlfcllem  13995
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-hom 13232  df-cco 13233  df-cat 13570  df-func 13732  df-nat 13817  df-fuc 13818
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