MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  catcfuccl Unicode version

Theorem catcfuccl 13941
Description: The category of categories for a weak universe is closed under the functor category operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
catcfuccl.c  |-  C  =  (CatCat `  U )
catcfuccl.b  |-  B  =  ( Base `  C
)
catcfuccl.o  |-  Q  =  ( X FuncCat  Y )
catcfuccl.u  |-  ( ph  ->  U  e. WUni )
catcfuccl.1  |-  ( ph  ->  om  e.  U )
catcfuccl.x  |-  ( ph  ->  X  e.  B )
catcfuccl.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
catcfuccl  |-  ( ph  ->  Q  e.  B )

Proof of Theorem catcfuccl
Dummy variables  a 
b  f  g  h  v  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catcfuccl.o . . . . 5  |-  Q  =  ( X FuncCat  Y )
2 eqid 2283 . . . . 5  |-  ( X 
Func  Y )  =  ( X  Func  Y )
3 eqid 2283 . . . . 5  |-  ( X Nat 
Y )  =  ( X Nat  Y )
4 eqid 2283 . . . . 5  |-  ( Base `  X )  =  (
Base `  X )
5 eqid 2283 . . . . 5  |-  (comp `  Y )  =  (comp `  Y )
6 inss2 3390 . . . . . 6  |-  ( U  i^i  Cat )  C_  Cat
7 catcfuccl.x . . . . . . 7  |-  ( ph  ->  X  e.  B )
8 catcfuccl.c . . . . . . . 8  |-  C  =  (CatCat `  U )
9 catcfuccl.b . . . . . . . 8  |-  B  =  ( Base `  C
)
10 catcfuccl.u . . . . . . . 8  |-  ( ph  ->  U  e. WUni )
118, 9, 10catcbas 13929 . . . . . . 7  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
127, 11eleqtrd 2359 . . . . . 6  |-  ( ph  ->  X  e.  ( U  i^i  Cat ) )
136, 12sseldi 3178 . . . . 5  |-  ( ph  ->  X  e.  Cat )
14 catcfuccl.y . . . . . . 7  |-  ( ph  ->  Y  e.  B )
1514, 11eleqtrd 2359 . . . . . 6  |-  ( ph  ->  Y  e.  ( U  i^i  Cat ) )
166, 15sseldi 3178 . . . . 5  |-  ( ph  ->  Y  e.  Cat )
17 eqidd 2284 . . . . 5  |-  ( ph  ->  ( v  e.  ( ( X  Func  Y
)  X.  ( X 
Func  Y ) ) ,  h  e.  ( X 
Func  Y )  |->  [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( X Nat  Y
) h ) ,  a  e.  ( f ( X Nat  Y ) g )  |->  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) )  =  ( v  e.  ( ( X  Func  Y
)  X.  ( X 
Func  Y ) ) ,  h  e.  ( X 
Func  Y )  |->  [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( X Nat  Y
) h ) ,  a  e.  ( f ( X Nat  Y ) g )  |->  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) ) )
181, 2, 3, 4, 5, 13, 16, 17fucval 13832 . . . 4  |-  ( ph  ->  Q  =  { <. (
Base `  ndx ) ,  ( X  Func  Y
) >. ,  <. (  Hom  `  ndx ) ,  ( X Nat  Y )
>. ,  <. (comp `  ndx ) ,  ( v  e.  ( ( X 
Func  Y )  X.  ( X  Func  Y ) ) ,  h  e.  ( X  Func  Y )  |-> 
[_ ( 1st `  v
)  /  f ]_ [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g ( X Nat  Y
) h ) ,  a  e.  ( f ( X Nat  Y ) g )  |->  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) ) >. } )
19 df-base 13153 . . . . . . 7  |-  Base  = Slot  1
20 catcfuccl.1 . . . . . . . 8  |-  ( ph  ->  om  e.  U )
2110, 20wunndx 13164 . . . . . . 7  |-  ( ph  ->  ndx  e.  U )
2219, 10, 21wunstr 13167 . . . . . 6  |-  ( ph  ->  ( Base `  ndx )  e.  U )
23 inss1 3389 . . . . . . . 8  |-  ( U  i^i  Cat )  C_  U
2423, 12sseldi 3178 . . . . . . 7  |-  ( ph  ->  X  e.  U )
2523, 15sseldi 3178 . . . . . . 7  |-  ( ph  ->  Y  e.  U )
2610, 24, 25wunfunc 13773 . . . . . 6  |-  ( ph  ->  ( X  Func  Y
)  e.  U )
2710, 22, 26wunop 8344 . . . . 5  |-  ( ph  -> 
<. ( Base `  ndx ) ,  ( X  Func  Y ) >.  e.  U
)
28 df-hom 13232 . . . . . . 7  |-  Hom  = Slot ; 1 4
2928, 10, 21wunstr 13167 . . . . . 6  |-  ( ph  ->  (  Hom  `  ndx )  e.  U )
3010, 24, 25wunnat 13830 . . . . . 6  |-  ( ph  ->  ( X Nat  Y )  e.  U )
3110, 29, 30wunop 8344 . . . . 5  |-  ( ph  -> 
<. (  Hom  `  ndx ) ,  ( X Nat  Y ) >.  e.  U
)
32 df-cco 13233 . . . . . . 7  |- comp  = Slot ; 1 5
3332, 10, 21wunstr 13167 . . . . . 6  |-  ( ph  ->  (comp `  ndx )  e.  U )
3410, 26, 26wunxp 8346 . . . . . . . 8  |-  ( ph  ->  ( ( X  Func  Y )  X.  ( X 
Func  Y ) )  e.  U )
3510, 34, 26wunxp 8346 . . . . . . 7  |-  ( ph  ->  ( ( ( X 
Func  Y )  X.  ( X  Func  Y ) )  X.  ( X  Func  Y ) )  e.  U
)
3632, 10, 25wunstr 13167 . . . . . . . . . . . . . 14  |-  ( ph  ->  (comp `  Y )  e.  U )
3710, 36wunrn 8351 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  (comp `  Y
)  e.  U )
3810, 37wununi 8328 . . . . . . . . . . . 12  |-  ( ph  ->  U. ran  (comp `  Y )  e.  U
)
3910, 38wunrn 8351 . . . . . . . . . . 11  |-  ( ph  ->  ran  U. ran  (comp `  Y )  e.  U
)
4010, 39wununi 8328 . . . . . . . . . 10  |-  ( ph  ->  U. ran  U. ran  (comp `  Y )  e.  U )
4110, 40wunpw 8329 . . . . . . . . 9  |-  ( ph  ->  ~P U. ran  U. ran  (comp `  Y )  e.  U )
4219, 10, 24wunstr 13167 . . . . . . . . 9  |-  ( ph  ->  ( Base `  X
)  e.  U )
4310, 41, 42wunmap 8348 . . . . . . . 8  |-  ( ph  ->  ( ~P U. ran  U.
ran  (comp `  Y )  ^m  ( Base `  X
) )  e.  U
)
4410, 30wunrn 8351 . . . . . . . . . 10  |-  ( ph  ->  ran  ( X Nat  Y
)  e.  U )
4510, 44wununi 8328 . . . . . . . . 9  |-  ( ph  ->  U. ran  ( X Nat 
Y )  e.  U
)
4610, 45, 45wunxp 8346 . . . . . . . 8  |-  ( ph  ->  ( U. ran  ( X Nat  Y )  X.  U. ran  ( X Nat  Y ) )  e.  U )
4710, 43, 46wunpm 8347 . . . . . . 7  |-  ( ph  ->  ( ( ~P U. ran  U. ran  (comp `  Y )  ^m  ( Base `  X ) ) 
^pm  ( U. ran  ( X Nat  Y )  X.  U. ran  ( X Nat 
Y ) ) )  e.  U )
48 fvex 5539 . . . . . . . . . . 11  |-  ( 1st `  v )  e.  _V
49 fvex 5539 . . . . . . . . . . . . . 14  |-  ( 2nd `  v )  e.  _V
50 ovex 5883 . . . . . . . . . . . . . . . . 17  |-  ( ~P
U. ran  U. ran  (comp `  Y )  ^m  ( Base `  X ) )  e.  _V
51 ovex 5883 . . . . . . . . . . . . . . . . . . . 20  |-  ( X Nat 
Y )  e.  _V
5251rnex 4942 . . . . . . . . . . . . . . . . . . 19  |-  ran  ( X Nat  Y )  e.  _V
5352uniex 4516 . . . . . . . . . . . . . . . . . 18  |-  U. ran  ( X Nat  Y )  e.  _V
5453, 53xpex 4801 . . . . . . . . . . . . . . . . 17  |-  ( U. ran  ( X Nat  Y )  X.  U. ran  ( X Nat  Y ) )  e. 
_V
55 eqid 2283 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) )  =  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) )
56 ovssunirn 5884 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( b `  x ) ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  Y )
( ( 1st `  h
) `  x )
) ( a `  x ) )  C_  U.
ran  ( <. (
( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) )
57 ovssunirn 5884 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( <.
( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) )  C_  U. ran  (comp `  Y )
58 rnss 4907 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) )  C_  U. ran  (comp `  Y )  ->  ran  ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  Y )
( ( 1st `  h
) `  x )
)  C_  ran  U. ran  (comp `  Y ) )
59 uniss 3848 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ran  ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  Y )
( ( 1st `  h
) `  x )
)  C_  ran  U. ran  (comp `  Y )  ->  U. ran  ( <. (
( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) )  C_  U. ran  U.
ran  (comp `  Y )
)
6057, 58, 59mp2b 9 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  U. ran  ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  Y )
( ( 1st `  h
) `  x )
)  C_  U. ran  U. ran  (comp `  Y )
6156, 60sstri 3188 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( b `  x ) ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  Y )
( ( 1st `  h
) `  x )
) ( a `  x ) )  C_  U.
ran  U. ran  (comp `  Y )
62 ovex 5883 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( b `  x ) ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  Y )
( ( 1st `  h
) `  x )
) ( a `  x ) )  e. 
_V
6362elpw 3631 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( b `  x
) ( <. (
( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) )  e.  ~P U. ran  U.
ran  (comp `  Y )  <->  ( ( b `  x
) ( <. (
( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) 
C_  U. ran  U. ran  (comp `  Y ) )
6461, 63mpbir 200 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( b `  x ) ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  Y )
( ( 1st `  h
) `  x )
) ( a `  x ) )  e. 
~P U. ran  U. ran  (comp `  Y )
6564a1i 10 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  e.  ( Base `  X
)  ->  ( (
b `  x )
( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  Y )
( ( 1st `  h
) `  x )
) ( a `  x ) )  e. 
~P U. ran  U. ran  (comp `  Y ) )
6655, 65fmpti 5683 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) : ( Base `  X ) --> ~P U. ran  U. ran  (comp `  Y )
67 fvex 5539 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  (comp `  Y )  e.  _V
6867rnex 4942 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ran  (comp `  Y )  e.  _V
6968uniex 4516 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  U. ran  (comp `  Y )  e. 
_V
7069rnex 4942 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ran  U. ran  (comp `  Y )  e.  _V
7170uniex 4516 . . . . . . . . . . . . . . . . . . . . . 22  |-  U. ran  U.
ran  (comp `  Y )  e.  _V
7271pwex 4193 . . . . . . . . . . . . . . . . . . . . 21  |-  ~P U. ran  U. ran  (comp `  Y )  e.  _V
73 fvex 5539 . . . . . . . . . . . . . . . . . . . . 21  |-  ( Base `  X )  e.  _V
7472, 73elmap 6796 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  ( Base `  X )  |->  ( ( b `  x ) ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  Y )
( ( 1st `  h
) `  x )
) ( a `  x ) ) )  e.  ( ~P U. ran  U. ran  (comp `  Y )  ^m  ( Base `  X ) )  <-> 
( x  e.  (
Base `  X )  |->  ( ( b `  x ) ( <.
( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) : ( Base `  X ) --> ~P U. ran  U. ran  (comp `  Y ) )
7566, 74mpbir 200 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) )  e.  ( ~P
U. ran  U. ran  (comp `  Y )  ^m  ( Base `  X ) )
7675rgen2w 2611 . . . . . . . . . . . . . . . . . 18  |-  A. b  e.  ( g ( X Nat 
Y ) h ) A. a  e.  ( f ( X Nat  Y
) g ) ( x  e.  ( Base `  X )  |->  ( ( b `  x ) ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  Y )
( ( 1st `  h
) `  x )
) ( a `  x ) ) )  e.  ( ~P U. ran  U. ran  (comp `  Y )  ^m  ( Base `  X ) )
77 eqid 2283 . . . . . . . . . . . . . . . . . . 19  |-  ( b  e.  ( g ( X Nat  Y ) h ) ,  a  e.  ( f ( X Nat 
Y ) g ) 
|->  ( x  e.  (
Base `  X )  |->  ( ( b `  x ) ( <.
( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )  =  ( b  e.  ( g ( X Nat  Y ) h ) ,  a  e.  ( f ( X Nat  Y ) g )  |->  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )
7877fmpt2 6191 . . . . . . . . . . . . . . . . . 18  |-  ( A. b  e.  ( g
( X Nat  Y ) h ) A. a  e.  ( f ( X Nat 
Y ) g ) ( x  e.  (
Base `  X )  |->  ( ( b `  x ) ( <.
( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) )  e.  ( ~P
U. ran  U. ran  (comp `  Y )  ^m  ( Base `  X ) )  <-> 
( b  e.  ( g ( X Nat  Y
) h ) ,  a  e.  ( f ( X Nat  Y ) g )  |->  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) : ( ( g ( X Nat 
Y ) h )  X.  ( f ( X Nat  Y ) g ) ) --> ( ~P
U. ran  U. ran  (comp `  Y )  ^m  ( Base `  X ) ) )
7976, 78mpbi 199 . . . . . . . . . . . . . . . . 17  |-  ( b  e.  ( g ( X Nat  Y ) h ) ,  a  e.  ( f ( X Nat 
Y ) g ) 
|->  ( x  e.  (
Base `  X )  |->  ( ( b `  x ) ( <.
( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) : ( ( g ( X Nat 
Y ) h )  X.  ( f ( X Nat  Y ) g ) ) --> ( ~P
U. ran  U. ran  (comp `  Y )  ^m  ( Base `  X ) )
80 ovssunirn 5884 . . . . . . . . . . . . . . . . . 18  |-  ( g ( X Nat  Y ) h )  C_  U. ran  ( X Nat  Y )
81 ovssunirn 5884 . . . . . . . . . . . . . . . . . 18  |-  ( f ( X Nat  Y ) g )  C_  U. ran  ( X Nat  Y )
82 xpss12 4792 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( g ( X Nat 
Y ) h ) 
C_  U. ran  ( X Nat 
Y )  /\  (
f ( X Nat  Y
) g )  C_  U.
ran  ( X Nat  Y
) )  ->  (
( g ( X Nat 
Y ) h )  X.  ( f ( X Nat  Y ) g ) )  C_  ( U. ran  ( X Nat  Y
)  X.  U. ran  ( X Nat  Y )
) )
8380, 81, 82mp2an 653 . . . . . . . . . . . . . . . . 17  |-  ( ( g ( X Nat  Y
) h )  X.  ( f ( X Nat 
Y ) g ) )  C_  ( U. ran  ( X Nat  Y )  X.  U. ran  ( X Nat  Y ) )
84 elpm2r 6788 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ~P U. ran  U. ran  (comp `  Y )  ^m  ( Base `  X ) )  e.  _V  /\  ( U. ran  ( X Nat  Y
)  X.  U. ran  ( X Nat  Y )
)  e.  _V )  /\  ( ( b  e.  ( g ( X Nat 
Y ) h ) ,  a  e.  ( f ( X Nat  Y
) g )  |->  ( x  e.  ( Base `  X )  |->  ( ( b `  x ) ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  Y )
( ( 1st `  h
) `  x )
) ( a `  x ) ) ) ) : ( ( g ( X Nat  Y
) h )  X.  ( f ( X Nat 
Y ) g ) ) --> ( ~P U. ran  U. ran  (comp `  Y )  ^m  ( Base `  X ) )  /\  ( ( g ( X Nat  Y ) h )  X.  (
f ( X Nat  Y
) g ) ) 
C_  ( U. ran  ( X Nat  Y )  X.  U. ran  ( X Nat 
Y ) ) ) )  ->  ( b  e.  ( g ( X Nat 
Y ) h ) ,  a  e.  ( f ( X Nat  Y
) g )  |->  ( x  e.  ( Base `  X )  |->  ( ( b `  x ) ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  Y )
( ( 1st `  h
) `  x )
) ( a `  x ) ) ) )  e.  ( ( ~P U. ran  U. ran  (comp `  Y )  ^m  ( Base `  X
) )  ^pm  ( U. ran  ( X Nat  Y
)  X.  U. ran  ( X Nat  Y )
) ) )
8550, 54, 79, 83, 84mp4an 654 . . . . . . . . . . . . . . . 16  |-  ( b  e.  ( g ( X Nat  Y ) h ) ,  a  e.  ( f ( X Nat 
Y ) g ) 
|->  ( x  e.  (
Base `  X )  |->  ( ( b `  x ) ( <.
( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )  e.  ( ( ~P U. ran  U.
ran  (comp `  Y )  ^m  ( Base `  X
) )  ^pm  ( U. ran  ( X Nat  Y
)  X.  U. ran  ( X Nat  Y )
) )
8685sbcth 3005 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  v )  e.  _V  ->  [. ( 2nd `  v )  / 
g ]. ( b  e.  ( g ( X Nat 
Y ) h ) ,  a  e.  ( f ( X Nat  Y
) g )  |->  ( x  e.  ( Base `  X )  |->  ( ( b `  x ) ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  Y )
( ( 1st `  h
) `  x )
) ( a `  x ) ) ) )  e.  ( ( ~P U. ran  U. ran  (comp `  Y )  ^m  ( Base `  X
) )  ^pm  ( U. ran  ( X Nat  Y
)  X.  U. ran  ( X Nat  Y )
) ) )
87 sbcel1g 3100 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  v )  e.  _V  ->  ( [. ( 2nd `  v
)  /  g ]. ( b  e.  ( g ( X Nat  Y
) h ) ,  a  e.  ( f ( X Nat  Y ) g )  |->  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )  e.  ( ( ~P U. ran  U.
ran  (comp `  Y )  ^m  ( Base `  X
) )  ^pm  ( U. ran  ( X Nat  Y
)  X.  U. ran  ( X Nat  Y )
) )  <->  [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( X Nat  Y
) h ) ,  a  e.  ( f ( X Nat  Y ) g )  |->  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )  e.  ( ( ~P U. ran  U.
ran  (comp `  Y )  ^m  ( Base `  X
) )  ^pm  ( U. ran  ( X Nat  Y
)  X.  U. ran  ( X Nat  Y )
) ) ) )
8886, 87mpbid 201 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  v )  e.  _V  ->  [_ ( 2nd `  v )  / 
g ]_ ( b  e.  ( g ( X Nat 
Y ) h ) ,  a  e.  ( f ( X Nat  Y
) g )  |->  ( x  e.  ( Base `  X )  |->  ( ( b `  x ) ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  Y )
( ( 1st `  h
) `  x )
) ( a `  x ) ) ) )  e.  ( ( ~P U. ran  U. ran  (comp `  Y )  ^m  ( Base `  X
) )  ^pm  ( U. ran  ( X Nat  Y
)  X.  U. ran  ( X Nat  Y )
) ) )
8949, 88ax-mp 8 . . . . . . . . . . . . 13  |-  [_ ( 2nd `  v )  / 
g ]_ ( b  e.  ( g ( X Nat 
Y ) h ) ,  a  e.  ( f ( X Nat  Y
) g )  |->  ( x  e.  ( Base `  X )  |->  ( ( b `  x ) ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  Y )
( ( 1st `  h
) `  x )
) ( a `  x ) ) ) )  e.  ( ( ~P U. ran  U. ran  (comp `  Y )  ^m  ( Base `  X
) )  ^pm  ( U. ran  ( X Nat  Y
)  X.  U. ran  ( X Nat  Y )
) )
9089sbcth 3005 . . . . . . . . . . . 12  |-  ( ( 1st `  v )  e.  _V  ->  [. ( 1st `  v )  / 
f ]. [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( X Nat  Y
) h ) ,  a  e.  ( f ( X Nat  Y ) g )  |->  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )  e.  ( ( ~P U. ran  U.
ran  (comp `  Y )  ^m  ( Base `  X
) )  ^pm  ( U. ran  ( X Nat  Y
)  X.  U. ran  ( X Nat  Y )
) ) )
91 sbcel1g 3100 . . . . . . . . . . . 12  |-  ( ( 1st `  v )  e.  _V  ->  ( [. ( 1st `  v
)  /  f ]. [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g ( X Nat  Y
) h ) ,  a  e.  ( f ( X Nat  Y ) g )  |->  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )  e.  ( ( ~P U. ran  U.
ran  (comp `  Y )  ^m  ( Base `  X
) )  ^pm  ( U. ran  ( X Nat  Y
)  X.  U. ran  ( X Nat  Y )
) )  <->  [_ ( 1st `  v )  /  f ]_ [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g ( X Nat  Y
) h ) ,  a  e.  ( f ( X Nat  Y ) g )  |->  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )  e.  ( ( ~P U. ran  U.
ran  (comp `  Y )  ^m  ( Base `  X
) )  ^pm  ( U. ran  ( X Nat  Y
)  X.  U. ran  ( X Nat  Y )
) ) ) )
9290, 91mpbid 201 . . . . . . . . . . 11  |-  ( ( 1st `  v )  e.  _V  ->  [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( X Nat  Y
) h ) ,  a  e.  ( f ( X Nat  Y ) g )  |->  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )  e.  ( ( ~P U. ran  U.
ran  (comp `  Y )  ^m  ( Base `  X
) )  ^pm  ( U. ran  ( X Nat  Y
)  X.  U. ran  ( X Nat  Y )
) ) )
9348, 92ax-mp 8 . . . . . . . . . 10  |-  [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( X Nat  Y
) h ) ,  a  e.  ( f ( X Nat  Y ) g )  |->  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )  e.  ( ( ~P U. ran  U.
ran  (comp `  Y )  ^m  ( Base `  X
) )  ^pm  ( U. ran  ( X Nat  Y
)  X.  U. ran  ( X Nat  Y )
) )
9493rgen2w 2611 . . . . . . . . 9  |-  A. v  e.  ( ( X  Func  Y )  X.  ( X 
Func  Y ) ) A. h  e.  ( X  Func  Y ) [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( X Nat  Y
) h ) ,  a  e.  ( f ( X Nat  Y ) g )  |->  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )  e.  ( ( ~P U. ran  U.
ran  (comp `  Y )  ^m  ( Base `  X
) )  ^pm  ( U. ran  ( X Nat  Y
)  X.  U. ran  ( X Nat  Y )
) )
95 eqid 2283 . . . . . . . . . 10  |-  ( v  e.  ( ( X 
Func  Y )  X.  ( X  Func  Y ) ) ,  h  e.  ( X  Func  Y )  |-> 
[_ ( 1st `  v
)  /  f ]_ [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g ( X Nat  Y
) h ) ,  a  e.  ( f ( X Nat  Y ) g )  |->  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) )  =  ( v  e.  ( ( X  Func  Y
)  X.  ( X 
Func  Y ) ) ,  h  e.  ( X 
Func  Y )  |->  [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( X Nat  Y
) h ) ,  a  e.  ( f ( X Nat  Y ) g )  |->  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) )
9695fmpt2 6191 . . . . . . . . 9  |-  ( A. v  e.  ( ( X  Func  Y )  X.  ( X  Func  Y
) ) A. h  e.  ( X  Func  Y
) [_ ( 1st `  v
)  /  f ]_ [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g ( X Nat  Y
) h ) ,  a  e.  ( f ( X Nat  Y ) g )  |->  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) )  e.  ( ( ~P U. ran  U.
ran  (comp `  Y )  ^m  ( Base `  X
) )  ^pm  ( U. ran  ( X Nat  Y
)  X.  U. ran  ( X Nat  Y )
) )  <->  ( v  e.  ( ( X  Func  Y )  X.  ( X 
Func  Y ) ) ,  h  e.  ( X 
Func  Y )  |->  [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( X Nat  Y
) h ) ,  a  e.  ( f ( X Nat  Y ) g )  |->  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) ) : ( ( ( X 
Func  Y )  X.  ( X  Func  Y ) )  X.  ( X  Func  Y ) ) --> ( ( ~P U. ran  U. ran  (comp `  Y )  ^m  ( Base `  X
) )  ^pm  ( U. ran  ( X Nat  Y
)  X.  U. ran  ( X Nat  Y )
) ) )
9794, 96mpbi 199 . . . . . . . 8  |-  ( v  e.  ( ( X 
Func  Y )  X.  ( X  Func  Y ) ) ,  h  e.  ( X  Func  Y )  |-> 
[_ ( 1st `  v
)  /  f ]_ [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g ( X Nat  Y
) h ) ,  a  e.  ( f ( X Nat  Y ) g )  |->  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) ) : ( ( ( X 
Func  Y )  X.  ( X  Func  Y ) )  X.  ( X  Func  Y ) ) --> ( ( ~P U. ran  U. ran  (comp `  Y )  ^m  ( Base `  X
) )  ^pm  ( U. ran  ( X Nat  Y
)  X.  U. ran  ( X Nat  Y )
) )
9897a1i 10 . . . . . . 7  |-  ( ph  ->  ( v  e.  ( ( X  Func  Y
)  X.  ( X 
Func  Y ) ) ,  h  e.  ( X 
Func  Y )  |->  [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( X Nat  Y
) h ) ,  a  e.  ( f ( X Nat  Y ) g )  |->  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) ) : ( ( ( X 
Func  Y )  X.  ( X  Func  Y ) )  X.  ( X  Func  Y ) ) --> ( ( ~P U. ran  U. ran  (comp `  Y )  ^m  ( Base `  X
) )  ^pm  ( U. ran  ( X Nat  Y
)  X.  U. ran  ( X Nat  Y )
) ) )
9910, 35, 47, 98wunf 8349 . . . . . 6  |-  ( ph  ->  ( v  e.  ( ( X  Func  Y
)  X.  ( X 
Func  Y ) ) ,  h  e.  ( X 
Func  Y )  |->  [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( X Nat  Y
) h ) ,  a  e.  ( f ( X Nat  Y ) g )  |->  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) )  e.  U )
10010, 33, 99wunop 8344 . . . . 5  |-  ( ph  -> 
<. (comp `  ndx ) ,  ( v  e.  ( ( X  Func  Y
)  X.  ( X 
Func  Y ) ) ,  h  e.  ( X 
Func  Y )  |->  [_ ( 1st `  v )  / 
f ]_ [_ ( 2nd `  v )  /  g ]_ ( b  e.  ( g ( X Nat  Y
) h ) ,  a  e.  ( f ( X Nat  Y ) g )  |->  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) ) >.  e.  U )
10110, 27, 31, 100wuntp 8333 . . . 4  |-  ( ph  ->  { <. ( Base `  ndx ) ,  ( X  Func  Y ) >. ,  <. (  Hom  `  ndx ) ,  ( X Nat  Y )
>. ,  <. (comp `  ndx ) ,  ( v  e.  ( ( X 
Func  Y )  X.  ( X  Func  Y ) ) ,  h  e.  ( X  Func  Y )  |-> 
[_ ( 1st `  v
)  /  f ]_ [_ ( 2nd `  v
)  /  g ]_ ( b  e.  ( g ( X Nat  Y
) h ) ,  a  e.  ( f ( X Nat  Y ) g )  |->  ( x  e.  ( Base `  X
)  |->  ( ( b `
 x ) (
<. ( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) ) ( a `
 x ) ) ) ) ) >. }  e.  U )
10218, 101eqeltrd 2357 . . 3  |-  ( ph  ->  Q  e.  U )
1031, 13, 16fuccat 13844 . . 3  |-  ( ph  ->  Q  e.  Cat )
104 elin 3358 . . 3  |-  ( Q  e.  ( U  i^i  Cat )  <->  ( Q  e.  U  /\  Q  e. 
Cat ) )
105102, 103, 104sylanbrc 645 . 2  |-  ( ph  ->  Q  e.  ( U  i^i  Cat ) )
106105, 11eleqtrrd 2360 1  |-  ( ph  ->  Q  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   [.wsbc 2991   [_csb 3081    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   {ctp 3642   <.cop 3643   U.cuni 3827    e. cmpt 4077   omcom 4656    X. cxp 4687   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121    ^m cmap 6772    ^pm cpm 6773  WUnicwun 8322   1c1 8738   4c4 9797   5c5 9798  ;cdc 10124   ndxcnx 13145   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566    Func cfunc 13728   Nat cnat 13815   FuncCat cfuc 13816  CatCatccatc 13926
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-inf2 7342  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-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-omul 6484  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-wun 8324  df-ni 8496  df-pli 8497  df-mi 8498  df-lti 8499  df-plpq 8532  df-mpq 8533  df-ltpq 8534  df-enq 8535  df-nq 8536  df-erq 8537  df-plq 8538  df-mq 8539  df-1nq 8540  df-rq 8541  df-ltnq 8542  df-np 8605  df-plp 8607  df-ltp 8609  df-enr 8681  df-nr 8682  df-c 8743  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-hom 13232  df-cco 13233  df-cat 13570  df-cid 13571  df-func 13732  df-nat 13817  df-fuc 13818  df-catc 13927
  Copyright terms: Public domain W3C validator