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Theorem catcfuccl 14980
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 2443 . . . . 5  |-  ( X 
Func  Y )  =  ( X  Func  Y )
3 eqid 2443 . . . . 5  |-  ( X Nat 
Y )  =  ( X Nat  Y )
4 eqid 2443 . . . . 5  |-  ( Base `  X )  =  (
Base `  X )
5 eqid 2443 . . . . 5  |-  (comp `  Y )  =  (comp `  Y )
6 inss2 3574 . . . . . 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 14968 . . . . . . 7  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
127, 11eleqtrd 2519 . . . . . 6  |-  ( ph  ->  X  e.  ( U  i^i  Cat ) )
136, 12sseldi 3357 . . . . 5  |-  ( ph  ->  X  e.  Cat )
14 catcfuccl.y . . . . . . 7  |-  ( ph  ->  Y  e.  B )
1514, 11eleqtrd 2519 . . . . . 6  |-  ( ph  ->  Y  e.  ( U  i^i  Cat ) )
166, 15sseldi 3357 . . . . 5  |-  ( ph  ->  Y  e.  Cat )
17 eqidd 2444 . . . . 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 14871 . . . 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 14182 . . . . . . 7  |-  Base  = Slot  1
20 catcfuccl.1 . . . . . . . 8  |-  ( ph  ->  om  e.  U )
2110, 20wunndx 14193 . . . . . . 7  |-  ( ph  ->  ndx  e.  U )
2219, 10, 21wunstr 14196 . . . . . 6  |-  ( ph  ->  ( Base `  ndx )  e.  U )
23 inss1 3573 . . . . . . . 8  |-  ( U  i^i  Cat )  C_  U
2423, 12sseldi 3357 . . . . . . 7  |-  ( ph  ->  X  e.  U )
2523, 15sseldi 3357 . . . . . . 7  |-  ( ph  ->  Y  e.  U )
2610, 24, 25wunfunc 14812 . . . . . 6  |-  ( ph  ->  ( X  Func  Y
)  e.  U )
2710, 22, 26wunop 8892 . . . . 5  |-  ( ph  -> 
<. ( Base `  ndx ) ,  ( X  Func  Y ) >.  e.  U
)
28 df-hom 14265 . . . . . . 7  |-  Hom  = Slot ; 1 4
2928, 10, 21wunstr 14196 . . . . . 6  |-  ( ph  ->  ( Hom  `  ndx )  e.  U )
3010, 24, 25wunnat 14869 . . . . . 6  |-  ( ph  ->  ( X Nat  Y )  e.  U )
3110, 29, 30wunop 8892 . . . . 5  |-  ( ph  -> 
<. ( Hom  `  ndx ) ,  ( X Nat  Y ) >.  e.  U
)
32 df-cco 14266 . . . . . . 7  |- comp  = Slot ; 1 5
3332, 10, 21wunstr 14196 . . . . . 6  |-  ( ph  ->  (comp `  ndx )  e.  U )
3410, 26, 26wunxp 8894 . . . . . . . 8  |-  ( ph  ->  ( ( X  Func  Y )  X.  ( X 
Func  Y ) )  e.  U )
3510, 34, 26wunxp 8894 . . . . . . 7  |-  ( ph  ->  ( ( ( X 
Func  Y )  X.  ( X  Func  Y ) )  X.  ( X  Func  Y ) )  e.  U
)
3632, 10, 25wunstr 14196 . . . . . . . . . . . . . 14  |-  ( ph  ->  (comp `  Y )  e.  U )
3710, 36wunrn 8899 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  (comp `  Y
)  e.  U )
3810, 37wununi 8876 . . . . . . . . . . . 12  |-  ( ph  ->  U. ran  (comp `  Y )  e.  U
)
3910, 38wunrn 8899 . . . . . . . . . . 11  |-  ( ph  ->  ran  U. ran  (comp `  Y )  e.  U
)
4010, 39wununi 8876 . . . . . . . . . 10  |-  ( ph  ->  U. ran  U. ran  (comp `  Y )  e.  U )
4110, 40wunpw 8877 . . . . . . . . 9  |-  ( ph  ->  ~P U. ran  U. ran  (comp `  Y )  e.  U )
4219, 10, 24wunstr 14196 . . . . . . . . 9  |-  ( ph  ->  ( Base `  X
)  e.  U )
4310, 41, 42wunmap 8896 . . . . . . . 8  |-  ( ph  ->  ( ~P U. ran  U.
ran  (comp `  Y )  ^m  ( Base `  X
) )  e.  U
)
4410, 30wunrn 8899 . . . . . . . . . 10  |-  ( ph  ->  ran  ( X Nat  Y
)  e.  U )
4510, 44wununi 8876 . . . . . . . . 9  |-  ( ph  ->  U. ran  ( X Nat 
Y )  e.  U
)
4610, 45, 45wunxp 8894 . . . . . . . 8  |-  ( ph  ->  ( U. ran  ( X Nat  Y )  X.  U. ran  ( X Nat  Y ) )  e.  U )
4710, 43, 46wunpm 8895 . . . . . . 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 5704 . . . . . . . . . . 11  |-  ( 1st `  v )  e.  _V
49 fvex 5704 . . . . . . . . . . . . . 14  |-  ( 2nd `  v )  e.  _V
50 ovex 6119 . . . . . . . . . . . . . . . . 17  |-  ( ~P
U. ran  U. ran  (comp `  Y )  ^m  ( Base `  X ) )  e.  _V
51 ovex 6119 . . . . . . . . . . . . . . . . . . . 20  |-  ( X Nat 
Y )  e.  _V
5251rnex 6515 . . . . . . . . . . . . . . . . . . 19  |-  ran  ( X Nat  Y )  e.  _V
5352uniex 6379 . . . . . . . . . . . . . . . . . 18  |-  U. ran  ( X Nat  Y )  e.  _V
5453, 53xpex 6511 . . . . . . . . . . . . . . . . 17  |-  ( U. ran  ( X Nat  Y )  X.  U. ran  ( X Nat  Y ) )  e. 
_V
55 eqid 2443 . . . . . . . . . . . . . . . . . . . . 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 6120 . . . . . . . . . . . . . . . . . . . . . . . 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 6120 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( <.
( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) )  C_  U. ran  (comp `  Y )
58 rnss 5071 . . . . . . . . . . . . . . . . . . . . . . . . 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 4115 . . . . . . . . . . . . . . . . . . . . . . . . 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 10 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  U. ran  ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  Y )
( ( 1st `  h
) `  x )
)  C_  U. ran  U. ran  (comp `  Y )
6156, 60sstri 3368 . . . . . . . . . . . . . . . . . . . . . . 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 6119 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( b `  x ) ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  Y )
( ( 1st `  h
) `  x )
) ( a `  x ) )  e. 
_V
6362elpw 3869 . . . . . . . . . . . . . . . . . . . . . . 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 209 . . . . . . . . . . . . . . . . . . . . . 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 11 . . . . . . . . . . . . . . . . . . . . 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 5869 . . . . . . . . . . . . . . . . . . . 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 5704 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  (comp `  Y )  e.  _V
6867rnex 6515 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ran  (comp `  Y )  e.  _V
6968uniex 6379 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  U. ran  (comp `  Y )  e. 
_V
7069rnex 6515 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ran  U. ran  (comp `  Y )  e.  _V
7170uniex 6379 . . . . . . . . . . . . . . . . . . . . . 22  |-  U. ran  U.
ran  (comp `  Y )  e.  _V
7271pwex 4478 . . . . . . . . . . . . . . . . . . . . 21  |-  ~P U. ran  U. ran  (comp `  Y )  e.  _V
73 fvex 5704 . . . . . . . . . . . . . . . . . . . . 21  |-  ( Base `  X )  e.  _V
7472, 73elmap 7244 . . . . . . . . . . . . . . . . . . . 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 209 . . . . . . . . . . . . . . . . . . 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 2787 . . . . . . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . . . . . . . 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 6644 . . . . . . . . . . . . . . . . . 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 208 . . . . . . . . . . . . . . . . 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 6120 . . . . . . . . . . . . . . . . . 18  |-  ( g ( X Nat  Y ) h )  C_  U. ran  ( X Nat  Y )
81 ovssunirn 6120 . . . . . . . . . . . . . . . . . 18  |-  ( f ( X Nat  Y ) g )  C_  U. ran  ( X Nat  Y )
82 xpss12 4948 . . . . . . . . . . . . . . . . . 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 672 . . . . . . . . . . . . . . . . 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 7233 . . . . . . . . . . . . . . . . 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 673 . . . . . . . . . . . . . . . 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 3204 . . . . . . . . . . . . . . 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 3684 . . . . . . . . . . . . . . 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 210 . . . . . . . . . . . . . 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 5 . . . . . . . . . . . . 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 3204 . . . . . . . . . . . 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 3684 . . . . . . . . . . . 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 210 . . . . . . . . . . 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 5 . . . . . . . . . 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 2787 . . . . . . . . 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 2443 . . . . . . . . . 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 6644 . . . . . . . . 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 208 . . . . . . . 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 11 . . . . . . 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 8897 . . . . . 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 8892 . . . . 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 8881 . . . 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 2517 . . 3  |-  ( ph  ->  Q  e.  U )
1031, 13, 16fuccat 14883 . . 3  |-  ( ph  ->  Q  e.  Cat )
104102, 103elind 3543 . 2  |-  ( ph  ->  Q  e.  ( U  i^i  Cat ) )
105104, 11eleqtrrd 2520 1  |-  ( ph  ->  Q  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   A.wral 2718   _Vcvv 2975   [.wsbc 3189   [_csb 3291    i^i cin 3330    C_ wss 3331   ~Pcpw 3863   {ctp 3884   <.cop 3886   U.cuni 4094    e. cmpt 4353    X. cxp 4841   ran crn 4844   -->wf 5417   ` cfv 5421  (class class class)co 6094    e. cmpt2 6096   omcom 6479   1stc1st 6578   2ndc2nd 6579    ^m cmap 7217    ^pm cpm 7218  WUnicwun 8870   1c1 9286   4c4 10376   5c5 10377  ;cdc 10758   ndxcnx 14174   Basecbs 14177   Hom chom 14252  compcco 14253   Catccat 14605    Func cfunc 14767   Nat cnat 14854   FuncCat cfuc 14855  CatCatccatc 14965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-inf2 7850  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-1o 6923  df-oadd 6927  df-omul 6928  df-er 7104  df-ec 7106  df-qs 7110  df-map 7219  df-pm 7220  df-ixp 7267  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-wun 8872  df-ni 9044  df-pli 9045  df-mi 9046  df-lti 9047  df-plpq 9080  df-mpq 9081  df-ltpq 9082  df-enq 9083  df-nq 9084  df-erq 9085  df-plq 9086  df-mq 9087  df-1nq 9088  df-rq 9089  df-ltnq 9090  df-np 9153  df-plp 9155  df-ltp 9157  df-enr 9229  df-nr 9230  df-c 9291  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-nn 10326  df-2 10383  df-3 10384  df-4 10385  df-5 10386  df-6 10387  df-7 10388  df-8 10389  df-9 10390  df-10 10391  df-n0 10583  df-z 10650  df-dec 10759  df-uz 10865  df-fz 11441  df-struct 14179  df-ndx 14180  df-slot 14181  df-base 14182  df-hom 14265  df-cco 14266  df-cat 14609  df-cid 14610  df-func 14771  df-nat 14856  df-fuc 14857  df-catc 14966
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator