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Theorem funcres 13770
Description: A functor restricted to a subcategory is a functor. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
funcres.f  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
funcres.h  |-  ( ph  ->  H  e.  (Subcat `  C ) )
Assertion
Ref Expression
funcres  |-  ( ph  ->  ( F  |`f  H )  e.  ( ( C  |`cat  H )  Func  D ) )

Proof of Theorem funcres
Dummy variables  f 
g  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funcres.f . . . 4  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
2 funcres.h . . . 4  |-  ( ph  ->  H  e.  (Subcat `  C ) )
31, 2resfval 13766 . . 3  |-  ( ph  ->  ( F  |`f  H )  =  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
)
43fveq2d 5529 . . . . 5  |-  ( ph  ->  ( 2nd `  ( F  |`f  H ) )  =  ( 2nd `  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
) )
5 fvex 5539 . . . . . . 7  |-  ( 1st `  F )  e.  _V
65resex 4995 . . . . . 6  |-  ( ( 1st `  F )  |`  dom  dom  H )  e.  _V
7 dmexg 4939 . . . . . . 7  |-  ( H  e.  (Subcat `  C
)  ->  dom  H  e. 
_V )
8 mptexg 5745 . . . . . . 7  |-  ( dom 
H  e.  _V  ->  ( z  e.  dom  H  |->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) )  e.  _V )
92, 7, 83syl 18 . . . . . 6  |-  ( ph  ->  ( z  e.  dom  H 
|->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) )  e.  _V )
10 op2ndg 6133 . . . . . 6  |-  ( ( ( ( 1st `  F
)  |`  dom  dom  H
)  e.  _V  /\  ( z  e.  dom  H 
|->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) )  e.  _V )  -> 
( 2nd `  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
)  =  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) )
116, 9, 10sylancr 644 . . . . 5  |-  ( ph  ->  ( 2nd `  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
)  =  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) )
124, 11eqtrd 2315 . . . 4  |-  ( ph  ->  ( 2nd `  ( F  |`f  H ) )  =  ( z  e.  dom  H 
|->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) ) )
1312opeq2d 3803 . . 3  |-  ( ph  -> 
<. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( 2nd `  ( F  |`f  H ) ) >.  =  <. ( ( 1st `  F )  |`  dom  dom  H ) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
)
143, 13eqtr4d 2318 . 2  |-  ( ph  ->  ( F  |`f  H )  =  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( 2nd `  ( F  |`f  H ) ) >.
)
15 eqid 2283 . . . 4  |-  ( Base `  ( C  |`cat  H )
)  =  ( Base `  ( C  |`cat  H )
)
16 eqid 2283 . . . 4  |-  ( Base `  D )  =  (
Base `  D )
17 eqid 2283 . . . 4  |-  (  Hom  `  ( C  |`cat  H )
)  =  (  Hom  `  ( C  |`cat  H )
)
18 eqid 2283 . . . 4  |-  (  Hom  `  D )  =  (  Hom  `  D )
19 eqid 2283 . . . 4  |-  ( Id
`  ( C  |`cat  H
) )  =  ( Id `  ( C  |`cat 
H ) )
20 eqid 2283 . . . 4  |-  ( Id
`  D )  =  ( Id `  D
)
21 eqid 2283 . . . 4  |-  (comp `  ( C  |`cat  H ) )  =  (comp `  ( C  |`cat  H ) )
22 eqid 2283 . . . 4  |-  (comp `  D )  =  (comp `  D )
23 eqid 2283 . . . . 5  |-  ( C  |`cat 
H )  =  ( C  |`cat  H )
2423, 2subccat 13722 . . . 4  |-  ( ph  ->  ( C  |`cat  H )  e.  Cat )
25 funcrcl 13737 . . . . . 6  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
261, 25syl 15 . . . . 5  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
2726simprd 449 . . . 4  |-  ( ph  ->  D  e.  Cat )
28 eqid 2283 . . . . . . 7  |-  ( Base `  C )  =  (
Base `  C )
29 relfunc 13736 . . . . . . . 8  |-  Rel  ( C  Func  D )
30 1st2ndbr 6169 . . . . . . . 8  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
3129, 1, 30sylancr 644 . . . . . . 7  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
3228, 16, 31funcf1 13740 . . . . . 6  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
33 eqidd 2284 . . . . . . . 8  |-  ( ph  ->  dom  dom  H  =  dom  dom  H )
342, 33subcfn 13715 . . . . . . 7  |-  ( ph  ->  H  Fn  ( dom 
dom  H  X.  dom  dom  H ) )
352, 34, 28subcss1 13716 . . . . . 6  |-  ( ph  ->  dom  dom  H  C_  ( Base `  C ) )
36 fssres 5408 . . . . . 6  |-  ( ( ( 1st `  F
) : ( Base `  C ) --> ( Base `  D )  /\  dom  dom 
H  C_  ( Base `  C ) )  -> 
( ( 1st `  F
)  |`  dom  dom  H
) : dom  dom  H --> ( Base `  D
) )
3732, 35, 36syl2anc 642 . . . . 5  |-  ( ph  ->  ( ( 1st `  F
)  |`  dom  dom  H
) : dom  dom  H --> ( Base `  D
) )
3826simpld 445 . . . . . . 7  |-  ( ph  ->  C  e.  Cat )
3923, 28, 38, 34, 35rescbas 13706 . . . . . 6  |-  ( ph  ->  dom  dom  H  =  ( Base `  ( C  |`cat  H ) ) )
4039feq2d 5380 . . . . 5  |-  ( ph  ->  ( ( ( 1st `  F )  |`  dom  dom  H ) : dom  dom  H --> ( Base `  D
)  <->  ( ( 1st `  F )  |`  dom  dom  H ) : ( Base `  ( C  |`cat  H )
) --> ( Base `  D
) ) )
4137, 40mpbid 201 . . . 4  |-  ( ph  ->  ( ( 1st `  F
)  |`  dom  dom  H
) : ( Base `  ( C  |`cat  H )
) --> ( Base `  D
) )
42 fvex 5539 . . . . . . 7  |-  ( ( 2nd `  F ) `
 z )  e. 
_V
4342resex 4995 . . . . . 6  |-  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) )  e.  _V
44 eqid 2283 . . . . . 6  |-  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) )  =  ( z  e.  dom  H 
|->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) )
4543, 44fnmpti 5372 . . . . 5  |-  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) )  Fn 
dom  H
4612eqcomd 2288 . . . . . 6  |-  ( ph  ->  ( z  e.  dom  H 
|->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) )  =  ( 2nd `  ( F  |`f  H ) ) )
47 fndm 5343 . . . . . . . 8  |-  ( H  Fn  ( dom  dom  H  X.  dom  dom  H
)  ->  dom  H  =  ( dom  dom  H  X.  dom  dom  H )
)
4834, 47syl 15 . . . . . . 7  |-  ( ph  ->  dom  H  =  ( dom  dom  H  X.  dom  dom  H ) )
4939, 39xpeq12d 4714 . . . . . . 7  |-  ( ph  ->  ( dom  dom  H  X.  dom  dom  H )  =  ( ( Base `  ( C  |`cat  H )
)  X.  ( Base `  ( C  |`cat  H )
) ) )
5048, 49eqtrd 2315 . . . . . 6  |-  ( ph  ->  dom  H  =  ( ( Base `  ( C  |`cat  H ) )  X.  ( Base `  ( C  |`cat  H ) ) ) )
5146, 50fneq12d 5337 . . . . 5  |-  ( ph  ->  ( ( z  e. 
dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) )  Fn 
dom  H  <->  ( 2nd `  ( F  |`f  H ) )  Fn  ( ( Base `  ( C  |`cat  H ) )  X.  ( Base `  ( C  |`cat  H ) ) ) ) )
5245, 51mpbii 202 . . . 4  |-  ( ph  ->  ( 2nd `  ( F  |`f  H ) )  Fn  ( ( Base `  ( C  |`cat  H ) )  X.  ( Base `  ( C  |`cat  H ) ) ) )
53 eqid 2283 . . . . . . . 8  |-  (  Hom  `  C )  =  (  Hom  `  C )
5431adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  -> 
( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
5535adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  ->  dom  dom  H  C_  ( Base `  C ) )
56 simprl 732 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  ->  x  e.  ( Base `  ( C  |`cat  H )
) )
5739adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  ->  dom  dom  H  =  (
Base `  ( C  |`cat  H ) ) )
5856, 57eleqtrrd 2360 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  ->  x  e.  dom  dom  H
)
5955, 58sseldd 3181 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  ->  x  e.  ( Base `  C ) )
60 simprr 733 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  -> 
y  e.  ( Base `  ( C  |`cat  H )
) )
6160, 57eleqtrrd 2360 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  -> 
y  e.  dom  dom  H )
6255, 61sseldd 3181 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  -> 
y  e.  ( Base `  C ) )
6328, 53, 18, 54, 59, 62funcf2 13742 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  -> 
( x ( 2nd `  F ) y ) : ( x (  Hom  `  C )
y ) --> ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
) )
642adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  ->  H  e.  (Subcat `  C
) )
6534adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  ->  H  Fn  ( dom  dom 
H  X.  dom  dom  H ) )
6664, 65, 53, 58, 61subcss2 13717 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  -> 
( x H y )  C_  ( x
(  Hom  `  C ) y ) )
67 fssres 5408 . . . . . . 7  |-  ( ( ( x ( 2nd `  F ) y ) : ( x (  Hom  `  C )
y ) --> ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
)  /\  ( x H y )  C_  ( x (  Hom  `  C ) y ) )  ->  ( (
x ( 2nd `  F
) y )  |`  ( x H y ) ) : ( x H y ) --> ( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  F ) `  y
) ) )
6863, 66, 67syl2anc 642 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  -> 
( ( x ( 2nd `  F ) y )  |`  (
x H y ) ) : ( x H y ) --> ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
) )
691adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  ->  F  e.  ( C  Func  D ) )
7069, 64, 65, 58, 61resf2nd 13769 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  -> 
( x ( 2nd `  ( F  |`f  H ) ) y )  =  ( ( x ( 2nd `  F
) y )  |`  ( x H y ) ) )
7170feq1d 5379 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  -> 
( ( x ( 2nd `  ( F  |`f  H ) ) y ) : ( x H y ) --> ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
)  <->  ( ( x ( 2nd `  F
) y )  |`  ( x H y ) ) : ( x H y ) --> ( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  F ) `  y
) ) ) )
7268, 71mpbird 223 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  -> 
( x ( 2nd `  ( F  |`f  H ) ) y ) : ( x H y ) --> ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
) )
7323, 28, 38, 34, 35reschom 13707 . . . . . . . 8  |-  ( ph  ->  H  =  (  Hom  `  ( C  |`cat  H )
) )
7473adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  ->  H  =  (  Hom  `  ( C  |`cat  H )
) )
7574oveqd 5875 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  -> 
( x H y )  =  ( x (  Hom  `  ( C  |`cat  H ) ) y ) )
76 fvres 5542 . . . . . . . . 9  |-  ( x  e.  dom  dom  H  ->  ( ( ( 1st `  F )  |`  dom  dom  H ) `  x )  =  ( ( 1st `  F ) `  x
) )
7758, 76syl 15 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  -> 
( ( ( 1st `  F )  |`  dom  dom  H ) `  x )  =  ( ( 1st `  F ) `  x
) )
78 fvres 5542 . . . . . . . . 9  |-  ( y  e.  dom  dom  H  ->  ( ( ( 1st `  F )  |`  dom  dom  H ) `  y )  =  ( ( 1st `  F ) `  y
) )
7961, 78syl 15 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  -> 
( ( ( 1st `  F )  |`  dom  dom  H ) `  y )  =  ( ( 1st `  F ) `  y
) )
8077, 79oveq12d 5876 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  -> 
( ( ( ( 1st `  F )  |`  dom  dom  H ) `  x ) (  Hom  `  D ) ( ( ( 1st `  F
)  |`  dom  dom  H
) `  y )
)  =  ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
) )
8180eqcomd 2288 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  -> 
( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  F ) `  y
) )  =  ( ( ( ( 1st `  F )  |`  dom  dom  H ) `  x ) (  Hom  `  D
) ( ( ( 1st `  F )  |`  dom  dom  H ) `  y ) ) )
8275, 81feq23d 5386 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  -> 
( ( x ( 2nd `  ( F  |`f  H ) ) y ) : ( x H y ) --> ( ( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
)  <->  ( x ( 2nd `  ( F  |`f  H ) ) y ) : ( x (  Hom  `  ( C  |`cat  H ) ) y ) --> ( ( ( ( 1st `  F
)  |`  dom  dom  H
) `  x )
(  Hom  `  D ) ( ( ( 1st `  F )  |`  dom  dom  H ) `  y ) ) ) )
8372, 82mpbid 201 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  -> 
( x ( 2nd `  ( F  |`f  H ) ) y ) : ( x (  Hom  `  ( C  |`cat  H ) ) y ) --> ( ( ( ( 1st `  F
)  |`  dom  dom  H
) `  x )
(  Hom  `  D ) ( ( ( 1st `  F )  |`  dom  dom  H ) `  y ) ) )
841adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  ->  F  e.  ( C  Func  D ) )
852adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  ->  H  e.  (Subcat `  C
) )
8634adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  ->  H  Fn  ( dom  dom 
H  X.  dom  dom  H ) )
8739eleq2d 2350 . . . . . . . 8  |-  ( ph  ->  ( x  e.  dom  dom 
H  <->  x  e.  ( Base `  ( C  |`cat  H
) ) ) )
8887biimpar 471 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  ->  x  e.  dom  dom  H
)
8984, 85, 86, 88, 88resf2nd 13769 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  -> 
( x ( 2nd `  ( F  |`f  H ) ) x )  =  ( ( x ( 2nd `  F
) x )  |`  ( x H x ) ) )
90 eqid 2283 . . . . . . . 8  |-  ( Id
`  C )  =  ( Id `  C
)
9123, 85, 86, 90, 88subcid 13721 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  -> 
( ( Id `  C ) `  x
)  =  ( ( Id `  ( C  |`cat 
H ) ) `  x ) )
9291eqcomd 2288 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  -> 
( ( Id `  ( C  |`cat  H ) ) `  x )  =  ( ( Id `  C
) `  x )
)
9389, 92fveq12d 5531 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  -> 
( ( x ( 2nd `  ( F  |`f  H ) ) x ) `  ( ( Id `  ( C  |`cat 
H ) ) `  x ) )  =  ( ( ( x ( 2nd `  F
) x )  |`  ( x H x ) ) `  (
( Id `  C
) `  x )
) )
9431adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  -> 
( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
9539, 35eqsstr3d 3213 . . . . . . . 8  |-  ( ph  ->  ( Base `  ( C  |`cat  H ) )  C_  ( Base `  C )
)
9695sselda 3180 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  ->  x  e.  ( Base `  C ) )
9728, 90, 20, 94, 96funcid 13744 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  -> 
( ( x ( 2nd `  F ) x ) `  (
( Id `  C
) `  x )
)  =  ( ( Id `  D ) `
 ( ( 1st `  F ) `  x
) ) )
9885, 86, 88, 90subcidcl 13718 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  -> 
( ( Id `  C ) `  x
)  e.  ( x H x ) )
99 fvres 5542 . . . . . . 7  |-  ( ( ( Id `  C
) `  x )  e.  ( x H x )  ->  ( (
( x ( 2nd `  F ) x )  |`  ( x H x ) ) `  (
( Id `  C
) `  x )
)  =  ( ( x ( 2nd `  F
) x ) `  ( ( Id `  C ) `  x
) ) )
10098, 99syl 15 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  -> 
( ( ( x ( 2nd `  F
) x )  |`  ( x H x ) ) `  (
( Id `  C
) `  x )
)  =  ( ( x ( 2nd `  F
) x ) `  ( ( Id `  C ) `  x
) ) )
10188, 76syl 15 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  -> 
( ( ( 1st `  F )  |`  dom  dom  H ) `  x )  =  ( ( 1st `  F ) `  x
) )
102101fveq2d 5529 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  -> 
( ( Id `  D ) `  (
( ( 1st `  F
)  |`  dom  dom  H
) `  x )
)  =  ( ( Id `  D ) `
 ( ( 1st `  F ) `  x
) ) )
10397, 100, 1023eqtr4d 2325 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  -> 
( ( ( x ( 2nd `  F
) x )  |`  ( x H x ) ) `  (
( Id `  C
) `  x )
)  =  ( ( Id `  D ) `
 ( ( ( 1st `  F )  |`  dom  dom  H ) `  x ) ) )
10493, 103eqtrd 2315 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  -> 
( ( x ( 2nd `  ( F  |`f  H ) ) x ) `  ( ( Id `  ( C  |`cat 
H ) ) `  x ) )  =  ( ( Id `  D ) `  (
( ( 1st `  F
)  |`  dom  dom  H
) `  x )
) )
10523ad2ant1 976 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  H  e.  (Subcat `  C ) )
106343ad2ant1 976 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  H  Fn  ( dom  dom  H  X.  dom  dom  H ) )
107 simp21 988 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  x  e.  ( Base `  ( C  |`cat  H ) ) )
108393ad2ant1 976 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  dom  dom  H  =  ( Base `  ( C  |`cat  H ) ) )
109107, 108eleqtrrd 2360 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  x  e.  dom  dom  H )
110 eqid 2283 . . . . . . . 8  |-  (comp `  C )  =  (comp `  C )
111 simp22 989 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  y  e.  ( Base `  ( C  |`cat  H ) ) )
112111, 108eleqtrrd 2360 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  y  e.  dom  dom  H )
113 simp23 990 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  z  e.  ( Base `  ( C  |`cat  H ) ) )
114113, 108eleqtrrd 2360 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  z  e.  dom  dom  H )
115 simp3l 983 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  f  e.  ( x (  Hom  `  ( C  |`cat  H )
) y ) )
116733ad2ant1 976 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  H  =  (  Hom  `  ( C  |`cat  H ) ) )
117116oveqd 5875 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( x H y )  =  ( x (  Hom  `  ( C  |`cat  H )
) y ) )
118115, 117eleqtrrd 2360 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  f  e.  ( x H y ) )
119 simp3r 984 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) )
120116oveqd 5875 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( y H z )  =  ( y (  Hom  `  ( C  |`cat  H )
) z ) )
121119, 120eleqtrrd 2360 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  g  e.  ( y H z ) )
122105, 106, 109, 110, 112, 114, 118, 121subccocl 13719 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( g
( <. x ,  y
>. (comp `  C )
z ) f )  e.  ( x H z ) )
123 fvres 5542 . . . . . . 7  |-  ( ( g ( <. x ,  y >. (comp `  C ) z ) f )  e.  ( x H z )  ->  ( ( ( x ( 2nd `  F
) z )  |`  ( x H z ) ) `  (
g ( <. x ,  y >. (comp `  C ) z ) f ) )  =  ( ( x ( 2nd `  F ) z ) `  (
g ( <. x ,  y >. (comp `  C ) z ) f ) ) )
124122, 123syl 15 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( (
( x ( 2nd `  F ) z )  |`  ( x H z ) ) `  (
g ( <. x ,  y >. (comp `  C ) z ) f ) )  =  ( ( x ( 2nd `  F ) z ) `  (
g ( <. x ,  y >. (comp `  C ) z ) f ) ) )
125313ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( 1st `  F ) ( C 
Func  D ) ( 2nd `  F ) )
126353ad2ant1 976 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  dom  dom  H  C_  ( Base `  C
) )
127126, 109sseldd 3181 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  x  e.  ( Base `  C )
)
128126, 112sseldd 3181 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  y  e.  ( Base `  C )
)
129126, 114sseldd 3181 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  z  e.  ( Base `  C )
)
130105, 106, 53, 109, 112subcss2 13717 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( x H y )  C_  ( x (  Hom  `  C ) y ) )
131130, 118sseldd 3181 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  f  e.  ( x (  Hom  `  C ) y ) )
132105, 106, 53, 112, 114subcss2 13717 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( y H z )  C_  ( y (  Hom  `  C ) z ) )
133132, 121sseldd 3181 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  g  e.  ( y (  Hom  `  C ) z ) )
13428, 53, 110, 22, 125, 127, 128, 129, 131, 133funcco 13745 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( (
x ( 2nd `  F
) z ) `  ( g ( <.
x ,  y >.
(comp `  C )
z ) f ) )  =  ( ( ( y ( 2nd `  F ) z ) `
 g ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( x ( 2nd `  F
) y ) `  f ) ) )
135124, 134eqtrd 2315 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( (
( x ( 2nd `  F ) z )  |`  ( x H z ) ) `  (
g ( <. x ,  y >. (comp `  C ) z ) f ) )  =  ( ( ( y ( 2nd `  F
) z ) `  g ) ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( x ( 2nd `  F
) y ) `  f ) ) )
13613ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  F  e.  ( C  Func  D ) )
137136, 105, 106, 109, 114resf2nd 13769 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( x
( 2nd `  ( F  |`f  H ) ) z )  =  ( ( x ( 2nd `  F
) z )  |`  ( x H z ) ) )
13823, 28, 38, 34, 35, 110rescco 13709 . . . . . . . . . 10  |-  ( ph  ->  (comp `  C )  =  (comp `  ( C  |`cat  H ) ) )
1391383ad2ant1 976 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  (comp `  C
)  =  (comp `  ( C  |`cat  H ) ) )
140139eqcomd 2288 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  (comp `  ( C  |`cat  H ) )  =  (comp `  C )
)
141140oveqd 5875 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( <. x ,  y >. (comp `  ( C  |`cat  H )
) z )  =  ( <. x ,  y
>. (comp `  C )
z ) )
142141oveqd 5875 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( g
( <. x ,  y
>. (comp `  ( C  |`cat  H ) ) z ) f )  =  ( g ( <. x ,  y >. (comp `  C ) z ) f ) )
143137, 142fveq12d 5531 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( (
x ( 2nd `  ( F  |`f  H ) ) z ) `  ( g ( <. x ,  y
>. (comp `  ( C  |`cat  H ) ) z ) f ) )  =  ( ( ( x ( 2nd `  F
) z )  |`  ( x H z ) ) `  (
g ( <. x ,  y >. (comp `  C ) z ) f ) ) )
144109, 76syl 15 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( (
( 1st `  F
)  |`  dom  dom  H
) `  x )  =  ( ( 1st `  F ) `  x
) )
145112, 78syl 15 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( (
( 1st `  F
)  |`  dom  dom  H
) `  y )  =  ( ( 1st `  F ) `  y
) )
146144, 145opeq12d 3804 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  <. ( ( ( 1st `  F
)  |`  dom  dom  H
) `  x ) ,  ( ( ( 1st `  F )  |`  dom  dom  H ) `  y ) >.  =  <. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. )
147 fvres 5542 . . . . . . . 8  |-  ( z  e.  dom  dom  H  ->  ( ( ( 1st `  F )  |`  dom  dom  H ) `  z )  =  ( ( 1st `  F ) `  z
) )
148114, 147syl 15 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( (
( 1st `  F
)  |`  dom  dom  H
) `  z )  =  ( ( 1st `  F ) `  z
) )
149146, 148oveq12d 5876 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( <. ( ( ( 1st `  F
)  |`  dom  dom  H
) `  x ) ,  ( ( ( 1st `  F )  |`  dom  dom  H ) `  y ) >. (comp `  D ) ( ( ( 1st `  F
)  |`  dom  dom  H
) `  z )
)  =  ( <.
( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) )
150136, 105, 106, 112, 114resf2nd 13769 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( y
( 2nd `  ( F  |`f  H ) ) z )  =  ( ( y ( 2nd `  F
) z )  |`  ( y H z ) ) )
151150fveq1d 5527 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( (
y ( 2nd `  ( F  |`f  H ) ) z ) `  g )  =  ( ( ( y ( 2nd `  F
) z )  |`  ( y H z ) ) `  g
) )
152 fvres 5542 . . . . . . . 8  |-  ( g  e.  ( y H z )  ->  (
( ( y ( 2nd `  F ) z )  |`  (
y H z ) ) `  g )  =  ( ( y ( 2nd `  F
) z ) `  g ) )
153121, 152syl 15 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( (
( y ( 2nd `  F ) z )  |`  ( y H z ) ) `  g
)  =  ( ( y ( 2nd `  F
) z ) `  g ) )
154151, 153eqtrd 2315 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( (
y ( 2nd `  ( F  |`f  H ) ) z ) `  g )  =  ( ( y ( 2nd `  F
) z ) `  g ) )
155136, 105, 106, 109, 112resf2nd 13769 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( x
( 2nd `  ( F  |`f  H ) ) y )  =  ( ( x ( 2nd `  F
) y )  |`  ( x H y ) ) )
156155fveq1d 5527 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( (
x ( 2nd `  ( F  |`f  H ) ) y ) `  f )  =  ( ( ( x ( 2nd `  F
) y )  |`  ( x H y ) ) `  f
) )
157 fvres 5542 . . . . . . . 8  |-  ( f  e.  ( x H y )  ->  (
( ( x ( 2nd `  F ) y )  |`  (
x H y ) ) `  f )  =  ( ( x ( 2nd `  F
) y ) `  f ) )
158118, 157syl 15 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( (
( x ( 2nd `  F ) y )  |`  ( x H y ) ) `  f
)  =  ( ( x ( 2nd `  F
) y ) `  f ) )
159156, 158eqtrd 2315 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( (
x ( 2nd `  ( F  |`f  H ) ) y ) `  f )  =  ( ( x ( 2nd `  F
) y ) `  f ) )
160149, 154, 159oveq123d 5879 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( (
( y ( 2nd `  ( F  |`f  H ) ) z ) `  g ) ( <. ( ( ( 1st `  F )  |`  dom  dom  H ) `  x ) ,  ( ( ( 1st `  F
)  |`  dom  dom  H
) `  y ) >. (comp `  D )
( ( ( 1st `  F )  |`  dom  dom  H ) `  z ) ) ( ( x ( 2nd `  ( F  |`f  H ) ) y ) `  f ) )  =  ( ( ( y ( 2nd `  F ) z ) `
 g ) (
<. ( ( 1st `  F
) `  x ) ,  ( ( 1st `  F ) `  y
) >. (comp `  D
) ( ( 1st `  F ) `  z
) ) ( ( x ( 2nd `  F
) y ) `  f ) ) )
161135, 143, 1603eqtr4d 2325 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
)  /\  z  e.  ( Base `  ( C  |`cat  H ) ) )  /\  ( f  e.  ( x (  Hom  `  ( C  |`cat  H ) ) y )  /\  g  e.  ( y (  Hom  `  ( C  |`cat  H )
) z ) ) )  ->  ( (
x ( 2nd `  ( F  |`f  H ) ) z ) `  ( g ( <. x ,  y
>. (comp `  ( C  |`cat  H ) ) z ) f ) )  =  ( ( ( y ( 2nd `  ( F  |`f  H ) ) z ) `  g ) ( <. ( ( ( 1st `  F )  |`  dom  dom  H ) `  x ) ,  ( ( ( 1st `  F
)  |`  dom  dom  H
) `  y ) >. (comp `  D )
( ( ( 1st `  F )  |`  dom  dom  H ) `  z ) ) ( ( x ( 2nd `  ( F  |`f  H ) ) y ) `  f ) ) )
16215, 16, 17, 18, 19, 20, 21, 22, 24, 27, 41, 52, 83, 104, 161isfuncd 13739 . . 3  |-  ( ph  ->  ( ( 1st `  F
)  |`  dom  dom  H
) ( ( C  |`cat 
H )  Func  D
) ( 2nd `  ( F  |`f  H ) ) )
163 df-br 4024 . . 3  |-  ( ( ( 1st `  F
)  |`  dom  dom  H
) ( ( C  |`cat 
H )  Func  D
) ( 2nd `  ( F  |`f  H ) )  <->  <. ( ( 1st `  F )  |`  dom  dom  H ) ,  ( 2nd `  ( F  |`f  H ) ) >.  e.  ( ( C  |`cat  H
)  Func  D )
)
164162, 163sylib 188 . 2  |-  ( ph  -> 
<. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( 2nd `  ( F  |`f  H ) ) >.  e.  ( ( C  |`cat  H
)  Func  D )
)
16514, 164eqeltrd 2357 1  |-  ( ph  ->  ( F  |`f  H )  e.  ( ( C  |`cat  H )  Func  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   <.cop 3643   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   dom cdm 4689    |` cres 4691   Rel wrel 4694    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566   Idccid 13567    |`cat cresc 13685  Subcatcsubc 13686    Func cfunc 13728    |`f cresf 13731
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-rmo 2551  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-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-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  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-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-hom 13232  df-cco 13233  df-cat 13570  df-cid 13571  df-homf 13572  df-ssc 13687  df-resc 13688  df-subc 13689  df-func 13732  df-resf 13735
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