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Theorem fuccocl 15180
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 2460 . . . 4  |-  ( Base `  C )  =  (
Base `  C )
4 eqid 2460 . . . 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 15178 . . 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 2460 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
10 eqid 2460 . . . . . 6  |-  ( Hom  `  D )  =  ( Hom  `  D )
112natrcl 15166 . . . . . . . . . . 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 459 . . . . . . . . 9  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
14 funcrcl 15079 . . . . . . . . 9  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
1513, 14syl 16 . . . . . . . 8  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
1615simprd 463 . . . . . . 7  |-  ( ph  ->  D  e.  Cat )
1716adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  D  e.  Cat )
18 relfunc 15078 . . . . . . . . 9  |-  Rel  ( C  Func  D )
19 1st2ndbr 6823 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
2018, 13, 19sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
213, 9, 20funcf1 15082 . . . . . . 7  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
2221ffvelrnda 6012 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
232natrcl 15166 . . . . . . . . . . 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 459 . . . . . . . . 9  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
26 1st2ndbr 6823 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 1st `  G ) ( C  Func  D )
( 2nd `  G
) )
2718, 25, 26sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( 1st `  G
) ( C  Func  D ) ( 2nd `  G
) )
283, 9, 27funcf1 15082 . . . . . . 7  |-  ( ph  ->  ( 1st `  G
) : ( Base `  C ) --> ( Base `  D ) )
2928ffvelrnda 6012 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  G ) `  x )  e.  (
Base `  D )
)
3024simprd 463 . . . . . . . . 9  |-  ( ph  ->  H  e.  ( C 
Func  D ) )
31 1st2ndbr 6823 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  D )  /\  H  e.  ( C  Func  D
) )  ->  ( 1st `  H ) ( C  Func  D )
( 2nd `  H
) )
3218, 30, 31sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( 1st `  H
) ( C  Func  D ) ( 2nd `  H
) )
333, 9, 32funcf1 15082 . . . . . . 7  |-  ( ph  ->  ( 1st `  H
) : ( Base `  C ) --> ( Base `  D ) )
3433ffvelrnda 6012 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  H ) `  x )  e.  (
Base `  D )
)
352, 6nat1st2nd 15167 . . . . . . . 8  |-  ( ph  ->  R  e.  ( <.
( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
3635adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  R  e.  ( <. ( 1st `  F
) ,  ( 2nd `  F ) >. N <. ( 1st `  G ) ,  ( 2nd `  G
) >. ) )
37 simpr 461 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  x  e.  ( Base `  C )
)
382, 36, 3, 10, 37natcl 15169 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( R `  x )  e.  ( ( ( 1st `  F
) `  x )
( Hom  `  D ) ( ( 1st `  G
) `  x )
) )
392, 7nat1st2nd 15167 . . . . . . . 8  |-  ( ph  ->  S  e.  ( <.
( 1st `  G
) ,  ( 2nd `  G ) >. N <. ( 1st `  H ) ,  ( 2nd `  H
) >. ) )
4039adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  S  e.  ( <. ( 1st `  G
) ,  ( 2nd `  G ) >. N <. ( 1st `  H ) ,  ( 2nd `  H
) >. ) )
412, 40, 3, 10, 37natcl 15169 . . . . . 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 14929 . . . . 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 2871 . . . 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 5867 . . . . 5  |-  ( Base `  C )  e.  _V
45 mptelixpg 7496 . . . . 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 5 . . . 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 212 . . 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 2548 . 2  |-  ( ph  ->  ( S ( <. F ,  G >.  .xb 
H ) R )  e.  X_ x  e.  (
Base `  C )
( ( ( 1st `  F ) `  x
) ( Hom  `  D
) ( ( 1st `  H ) `  x
) ) )
4916adantr 465 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  D  e.  Cat )
5021adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( 1st `  F ) : (
Base `  C ) --> ( Base `  D )
)
51 simpr1 997 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  x  e.  ( Base `  C )
)
5250, 51ffvelrnd 6013 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
53 simpr2 998 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  y  e.  ( Base `  C )
)
5450, 53ffvelrnd 6013 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( ( 1st `  F ) `  y )  e.  (
Base `  D )
)
5528adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( 1st `  G ) : (
Base `  C ) --> ( Base `  D )
)
5655, 53ffvelrnd 6013 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( ( 1st `  G ) `  y )  e.  (
Base `  D )
)
57 eqid 2460 . . . . . . . 8  |-  ( Hom  `  C )  =  ( Hom  `  C )
5820adantr 465 . . . . . . . 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 15084 . . . . . . 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 999 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  f  e.  ( x ( Hom  `  C ) y ) )
6159, 60ffvelrnd 6013 . . . . . 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 465 . . . . . . 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 15169 . . . . . 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 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( 1st `  H ) : (
Base `  C ) --> ( Base `  D )
)
6564, 53ffvelrnd 6013 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  ( ( 1st `  H ) `  y )  e.  (
Base `  D )
)
6639adantr 465 . . . . . . 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 15169 . . . . . 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 14930 . . . . 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 15171 . . . . . . 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 6291 . . . . . 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 6013 . . . . . . 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 15169 . . . . . . 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 465 . . . . . . . . 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 15084 . . . . . . . 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 6013 . . . . . . 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 14930 . . . . . 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 15171 . . . . . . 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 6290 . . . . . 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 2507 . . . . 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 6013 . . . . . 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 15169 . . . . . 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 465 . . . . . . . 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 15084 . . . . . . 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 6013 . . . . . 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 14930 . . . . 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 2505 . . . 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 465 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  f  e.  (
x ( Hom  `  C
) y ) ) )  ->  R  e.  ( F N G ) )
887adantr 465 . . . . . 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 15179 . . . . 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 6290 . . . 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 15179 . . . . 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 6291 . . . 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 2511 . . 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 2879 . 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 15164 . 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 915 1  |-  ( ph  ->  ( S ( <. F ,  G >.  .xb 
H ) R )  e.  ( F N H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2807   _Vcvv 3106   <.cop 4026   class class class wbr 4440    |-> cmpt 4498   Rel wrel 4997   -->wf 5575   ` cfv 5579  (class class class)co 6275   1stc1st 6772   2ndc2nd 6773   X_cixp 7459   Basecbs 14479   Hom chom 14555  compcco 14556   Catccat 14908    Func cfunc 15070   Nat cnat 15157   FuncCat cfuc 15158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-fz 11662  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-hom 14568  df-cco 14569  df-cat 14912  df-func 15074  df-nat 15159  df-fuc 15160
This theorem is referenced by:  fucass  15184  fuccatid  15185  evlfcllem  15337  yonedalem3b  15395
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