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Theorem funcres 14020
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 14016 . . 3  |-  ( ph  ->  ( F  |`f  H )  =  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
)
43fveq2d 5672 . . . . 5  |-  ( ph  ->  ( 2nd `  ( F  |`f  H ) )  =  ( 2nd `  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) ) >.
) )
5 fvex 5682 . . . . . . 7  |-  ( 1st `  F )  e.  _V
65resex 5126 . . . . . 6  |-  ( ( 1st `  F )  |`  dom  dom  H )  e.  _V
7 dmexg 5070 . . . . . . 7  |-  ( H  e.  (Subcat `  C
)  ->  dom  H  e. 
_V )
8 mptexg 5904 . . . . . . 7  |-  ( dom 
H  e.  _V  ->  ( z  e.  dom  H  |->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) )  e.  _V )
92, 7, 83syl 19 . . . . . 6  |-  ( ph  ->  ( z  e.  dom  H 
|->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) )  e.  _V )
10 op2ndg 6299 . . . . . 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 645 . . . . 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 2419 . . . 4  |-  ( ph  ->  ( 2nd `  ( F  |`f  H ) )  =  ( z  e.  dom  H 
|->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) ) )
1312opeq2d 3933 . . 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 2422 . 2  |-  ( ph  ->  ( F  |`f  H )  =  <. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( 2nd `  ( F  |`f  H ) ) >.
)
15 eqid 2387 . . . 4  |-  ( Base `  ( C  |`cat  H )
)  =  ( Base `  ( C  |`cat  H )
)
16 eqid 2387 . . . 4  |-  ( Base `  D )  =  (
Base `  D )
17 eqid 2387 . . . 4  |-  (  Hom  `  ( C  |`cat  H )
)  =  (  Hom  `  ( C  |`cat  H )
)
18 eqid 2387 . . . 4  |-  (  Hom  `  D )  =  (  Hom  `  D )
19 eqid 2387 . . . 4  |-  ( Id
`  ( C  |`cat  H
) )  =  ( Id `  ( C  |`cat 
H ) )
20 eqid 2387 . . . 4  |-  ( Id
`  D )  =  ( Id `  D
)
21 eqid 2387 . . . 4  |-  (comp `  ( C  |`cat  H ) )  =  (comp `  ( C  |`cat  H ) )
22 eqid 2387 . . . 4  |-  (comp `  D )  =  (comp `  D )
23 eqid 2387 . . . . 5  |-  ( C  |`cat 
H )  =  ( C  |`cat  H )
2423, 2subccat 13972 . . . 4  |-  ( ph  ->  ( C  |`cat  H )  e.  Cat )
25 funcrcl 13987 . . . . . 6  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
261, 25syl 16 . . . . 5  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
2726simprd 450 . . . 4  |-  ( ph  ->  D  e.  Cat )
28 eqid 2387 . . . . . . 7  |-  ( Base `  C )  =  (
Base `  C )
29 relfunc 13986 . . . . . . . 8  |-  Rel  ( C  Func  D )
30 1st2ndbr 6335 . . . . . . . 8  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
3129, 1, 30sylancr 645 . . . . . . 7  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
3228, 16, 31funcf1 13990 . . . . . 6  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
33 eqidd 2388 . . . . . . . 8  |-  ( ph  ->  dom  dom  H  =  dom  dom  H )
342, 33subcfn 13965 . . . . . . 7  |-  ( ph  ->  H  Fn  ( dom 
dom  H  X.  dom  dom  H ) )
352, 34, 28subcss1 13966 . . . . . 6  |-  ( ph  ->  dom  dom  H  C_  ( Base `  C ) )
36 fssres 5550 . . . . . 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 643 . . . . 5  |-  ( ph  ->  ( ( 1st `  F
)  |`  dom  dom  H
) : dom  dom  H --> ( Base `  D
) )
3826simpld 446 . . . . . . 7  |-  ( ph  ->  C  e.  Cat )
3923, 28, 38, 34, 35rescbas 13956 . . . . . 6  |-  ( ph  ->  dom  dom  H  =  ( Base `  ( C  |`cat  H ) ) )
4039feq2d 5521 . . . . 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 202 . . . 4  |-  ( ph  ->  ( ( 1st `  F
)  |`  dom  dom  H
) : ( Base `  ( C  |`cat  H )
) --> ( Base `  D
) )
42 fvex 5682 . . . . . . 7  |-  ( ( 2nd `  F ) `
 z )  e. 
_V
4342resex 5126 . . . . . 6  |-  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) )  e.  _V
44 eqid 2387 . . . . . 6  |-  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) )  =  ( z  e.  dom  H 
|->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) )
4543, 44fnmpti 5513 . . . . 5  |-  ( z  e.  dom  H  |->  ( ( ( 2nd `  F
) `  z )  |`  ( H `  z
) ) )  Fn 
dom  H
4612eqcomd 2392 . . . . . 6  |-  ( ph  ->  ( z  e.  dom  H 
|->  ( ( ( 2nd `  F ) `  z
)  |`  ( H `  z ) ) )  =  ( 2nd `  ( F  |`f  H ) ) )
47 fndm 5484 . . . . . . . 8  |-  ( H  Fn  ( dom  dom  H  X.  dom  dom  H
)  ->  dom  H  =  ( dom  dom  H  X.  dom  dom  H )
)
4834, 47syl 16 . . . . . . 7  |-  ( ph  ->  dom  H  =  ( dom  dom  H  X.  dom  dom  H ) )
4939, 39xpeq12d 4843 . . . . . . 7  |-  ( ph  ->  ( dom  dom  H  X.  dom  dom  H )  =  ( ( Base `  ( C  |`cat  H )
)  X.  ( Base `  ( C  |`cat  H )
) ) )
5048, 49eqtrd 2419 . . . . . 6  |-  ( ph  ->  dom  H  =  ( ( Base `  ( C  |`cat  H ) )  X.  ( Base `  ( C  |`cat  H ) ) ) )
5146, 50fneq12d 5478 . . . . 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 203 . . . 4  |-  ( ph  ->  ( 2nd `  ( F  |`f  H ) )  Fn  ( ( Base `  ( C  |`cat  H ) )  X.  ( Base `  ( C  |`cat  H ) ) ) )
53 eqid 2387 . . . . . . . 8  |-  (  Hom  `  C )  =  (  Hom  `  C )
5431adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  -> 
( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
5535adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  ->  dom  dom  H  C_  ( Base `  C ) )
56 simprl 733 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  ->  x  e.  ( Base `  ( C  |`cat  H )
) )
5739adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  ->  dom  dom  H  =  (
Base `  ( C  |`cat  H ) ) )
5856, 57eleqtrrd 2464 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  ->  x  e.  dom  dom  H
)
5955, 58sseldd 3292 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  ->  x  e.  ( Base `  C ) )
60 simprr 734 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  -> 
y  e.  ( Base `  ( C  |`cat  H )
) )
6160, 57eleqtrrd 2464 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  -> 
y  e.  dom  dom  H )
6255, 61sseldd 3292 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  -> 
y  e.  ( Base `  C ) )
6328, 53, 18, 54, 59, 62funcf2 13992 . . . . . . 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 452 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  ->  H  e.  (Subcat `  C
) )
6534adantr 452 . . . . . . . 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 13967 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  -> 
( x H y )  C_  ( x
(  Hom  `  C ) y ) )
67 fssres 5550 . . . . . . 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 643 . . . . . 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 452 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  ->  F  e.  ( C  Func  D ) )
7069, 64, 65, 58, 61resf2nd 14019 . . . . . . 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 5520 . . . . . 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 224 . . . . 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 13957 . . . . . . . 8  |-  ( ph  ->  H  =  (  Hom  `  ( C  |`cat  H )
) )
7473adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  ( C  |`cat  H ) )  /\  y  e.  ( Base `  ( C  |`cat  H )
) ) )  ->  H  =  (  Hom  `  ( C  |`cat  H )
) )
7574oveqd 6037 . . . . . 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 5685 . . . . . . . . 9  |-  ( x  e.  dom  dom  H  ->  ( ( ( 1st `  F )  |`  dom  dom  H ) `  x )  =  ( ( 1st `  F ) `  x
) )
7758, 76syl 16 . . . . . . . 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 5685 . . . . . . . . 9  |-  ( y  e.  dom  dom  H  ->  ( ( ( 1st `  F )  |`  dom  dom  H ) `  y )  =  ( ( 1st `  F ) `  y
) )
7961, 78syl 16 . . . . . . . 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 6038 . . . . . . 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 2392 . . . . . 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 5528 . . . . 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 202 . . . 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 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  ->  F  e.  ( C  Func  D ) )
852adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  ->  H  e.  (Subcat `  C
) )
8634adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  ->  H  Fn  ( dom  dom 
H  X.  dom  dom  H ) )
8739eleq2d 2454 . . . . . . . 8  |-  ( ph  ->  ( x  e.  dom  dom 
H  <->  x  e.  ( Base `  ( C  |`cat  H
) ) ) )
8887biimpar 472 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  ->  x  e.  dom  dom  H
)
8984, 85, 86, 88, 88resf2nd 14019 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  -> 
( x ( 2nd `  ( F  |`f  H ) ) x )  =  ( ( x ( 2nd `  F
) x )  |`  ( x H x ) ) )
90 eqid 2387 . . . . . . . 8  |-  ( Id
`  C )  =  ( Id `  C
)
9123, 85, 86, 90, 88subcid 13971 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  -> 
( ( Id `  C ) `  x
)  =  ( ( Id `  ( C  |`cat 
H ) ) `  x ) )
9291eqcomd 2392 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  -> 
( ( Id `  ( C  |`cat  H ) ) `  x )  =  ( ( Id `  C
) `  x )
)
9389, 92fveq12d 5674 . . . . 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 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  -> 
( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
9539, 35eqsstr3d 3326 . . . . . . . 8  |-  ( ph  ->  ( Base `  ( C  |`cat  H ) )  C_  ( Base `  C )
)
9695sselda 3291 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  ->  x  e.  ( Base `  C ) )
9728, 90, 20, 94, 96funcid 13994 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  -> 
( ( x ( 2nd `  F ) x ) `  (
( Id `  C
) `  x )
)  =  ( ( Id `  D ) `
 ( ( 1st `  F ) `  x
) ) )
9885, 86, 88, 90subcidcl 13968 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  -> 
( ( Id `  C ) `  x
)  e.  ( x H x ) )
99 fvres 5685 . . . . . . 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 16 . . . . . 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 16 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  -> 
( ( ( 1st `  F )  |`  dom  dom  H ) `  x )  =  ( ( 1st `  F ) `  x
) )
102101fveq2d 5672 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  ( C  |`cat  H ) ) )  -> 
( ( Id `  D ) `  (
( ( 1st `  F
)  |`  dom  dom  H
) `  x )
)  =  ( ( Id `  D ) `
 ( ( 1st `  F ) `  x
) ) )
10397, 100, 1023eqtr4d 2429 . . . . 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 2419 . . . 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 978 . . . . . . . 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 978 . . . . . . . 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 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 ) ) )  ->  x  e.  ( Base `  ( C  |`cat  H ) ) )
108393ad2ant1 978 . . . . . . . . 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 2464 . . . . . . . 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 2387 . . . . . . . 8  |-  (comp `  C )  =  (comp `  C )
111 simp22 991 . . . . . . . . 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 2464 . . . . . . . 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 992 . . . . . . . . 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 2464 . . . . . . . 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 985 . . . . . . . . 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 978 . . . . . . . . . 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 6037 . . . . . . . . 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 2464 . . . . . . . 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 986 . . . . . . . . 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 6037 . . . . . . . . 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 2464 . . . . . . . 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 13969 . . . . . . 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 5685 . . . . . . 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 16 . . . . . 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 978 . . . . . . 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 978 . . . . . . . 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 3292 . . . . . . 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 3292 . . . . . . 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 3292 . . . . . . 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 13967 . . . . . . . 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 3292 . . . . . . 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 13967 . . . . . . . 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 3292 . . . . . . 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 13995 . . . . . 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 2419 . . . . 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 978 . . . . . . 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 14019 . . . . . 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 13959 . . . . . . . . . 10  |-  ( ph  ->  (comp `  C )  =  (comp `  ( C  |`cat  H ) ) )
1391383ad2ant1 978 . . . . . . . . 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 2392 . . . . . . . 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 6037 . . . . . . 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 6037 . . . . . 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 5674 . . . . 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 16 . . . . . . . 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 16 . . . . . . . 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 3934 . . . . . . 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 5685 . . . . . . . 8  |-  ( z  e.  dom  dom  H  ->  ( ( ( 1st `  F )  |`  dom  dom  H ) `  z )  =  ( ( 1st `  F ) `  z
) )
148114, 147syl 16 . . . . . . 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 6038 . . . . . 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 14019 . . . . . . . 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 5670 . . . . . . 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 5685 . . . . . . . 8  |-  ( g  e.  ( y H z )  ->  (
( ( y ( 2nd `  F ) z )  |`  (
y H z ) ) `  g )  =  ( ( y ( 2nd `  F
) z ) `  g ) )
153121, 152syl 16 . . . . . . 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 2419 . . . . . 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 14019 . . . . . . . 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 5670 . . . . . . 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 5685 . . . . . . . 8  |-  ( f  e.  ( x H y )  ->  (
( ( x ( 2nd `  F ) y )  |`  (
x H y ) ) `  f )  =  ( ( x ( 2nd `  F
) y ) `  f ) )
158118, 157syl 16 . . . . . . 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 2419 . . . . . 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 6041 . . . . 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 2429 . . . 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 13989 . . 3  |-  ( ph  ->  ( ( 1st `  F
)  |`  dom  dom  H
) ( ( C  |`cat 
H )  Func  D
) ( 2nd `  ( F  |`f  H ) ) )
163 df-br 4154 . . 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 189 . 2  |-  ( ph  -> 
<. ( ( 1st `  F
)  |`  dom  dom  H
) ,  ( 2nd `  ( F  |`f  H ) ) >.  e.  ( ( C  |`cat  H
)  Func  D )
)
16514, 164eqeltrd 2461 1  |-  ( ph  ->  ( F  |`f  H )  e.  ( ( C  |`cat  H )  Func  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   _Vcvv 2899    C_ wss 3263   <.cop 3760   class class class wbr 4153    e. cmpt 4207    X. cxp 4816   dom cdm 4818    |` cres 4820   Rel wrel 4823    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020   1stc1st 6286   2ndc2nd 6287   Basecbs 13396    Hom chom 13467  compcco 13468   Catccat 13816   Idccid 13817    |`cat cresc 13935  Subcatcsubc 13936    Func cfunc 13978    |`f cresf 13981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-map 6956  df-pm 6957  df-ixp 7000  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-hom 13480  df-cco 13481  df-cat 13820  df-cid 13821  df-homf 13822  df-ssc 13937  df-resc 13938  df-subc 13939  df-func 13982  df-resf 13985
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