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Theorem catcfuccl 15306
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 2467 . . . . 5  |-  ( X 
Func  Y )  =  ( X  Func  Y )
3 eqid 2467 . . . . 5  |-  ( X Nat 
Y )  =  ( X Nat  Y )
4 eqid 2467 . . . . 5  |-  ( Base `  X )  =  (
Base `  X )
5 eqid 2467 . . . . 5  |-  (comp `  Y )  =  (comp `  Y )
6 inss2 3724 . . . . . 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 15294 . . . . . . 7  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
127, 11eleqtrd 2557 . . . . . 6  |-  ( ph  ->  X  e.  ( U  i^i  Cat ) )
136, 12sseldi 3507 . . . . 5  |-  ( ph  ->  X  e.  Cat )
14 catcfuccl.y . . . . . . 7  |-  ( ph  ->  Y  e.  B )
1514, 11eleqtrd 2557 . . . . . 6  |-  ( ph  ->  Y  e.  ( U  i^i  Cat ) )
166, 15sseldi 3507 . . . . 5  |-  ( ph  ->  Y  e.  Cat )
17 eqidd 2468 . . . . 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 15197 . . . 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 14507 . . . . . . 7  |-  Base  = Slot  1
20 catcfuccl.1 . . . . . . . 8  |-  ( ph  ->  om  e.  U )
2110, 20wunndx 14518 . . . . . . 7  |-  ( ph  ->  ndx  e.  U )
2219, 10, 21wunstr 14521 . . . . . 6  |-  ( ph  ->  ( Base `  ndx )  e.  U )
23 inss1 3723 . . . . . . . 8  |-  ( U  i^i  Cat )  C_  U
2423, 12sseldi 3507 . . . . . . 7  |-  ( ph  ->  X  e.  U )
2523, 15sseldi 3507 . . . . . . 7  |-  ( ph  ->  Y  e.  U )
2610, 24, 25wunfunc 15138 . . . . . 6  |-  ( ph  ->  ( X  Func  Y
)  e.  U )
2710, 22, 26wunop 9110 . . . . 5  |-  ( ph  -> 
<. ( Base `  ndx ) ,  ( X  Func  Y ) >.  e.  U
)
28 df-hom 14591 . . . . . . 7  |-  Hom  = Slot ; 1 4
2928, 10, 21wunstr 14521 . . . . . 6  |-  ( ph  ->  ( Hom  `  ndx )  e.  U )
3010, 24, 25wunnat 15195 . . . . . 6  |-  ( ph  ->  ( X Nat  Y )  e.  U )
3110, 29, 30wunop 9110 . . . . 5  |-  ( ph  -> 
<. ( Hom  `  ndx ) ,  ( X Nat  Y ) >.  e.  U
)
32 df-cco 14592 . . . . . . 7  |- comp  = Slot ; 1 5
3332, 10, 21wunstr 14521 . . . . . 6  |-  ( ph  ->  (comp `  ndx )  e.  U )
3410, 26, 26wunxp 9112 . . . . . . . 8  |-  ( ph  ->  ( ( X  Func  Y )  X.  ( X 
Func  Y ) )  e.  U )
3510, 34, 26wunxp 9112 . . . . . . 7  |-  ( ph  ->  ( ( ( X 
Func  Y )  X.  ( X  Func  Y ) )  X.  ( X  Func  Y ) )  e.  U
)
3632, 10, 25wunstr 14521 . . . . . . . . . . . . . 14  |-  ( ph  ->  (comp `  Y )  e.  U )
3710, 36wunrn 9117 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  (comp `  Y
)  e.  U )
3810, 37wununi 9094 . . . . . . . . . . . 12  |-  ( ph  ->  U. ran  (comp `  Y )  e.  U
)
3910, 38wunrn 9117 . . . . . . . . . . 11  |-  ( ph  ->  ran  U. ran  (comp `  Y )  e.  U
)
4010, 39wununi 9094 . . . . . . . . . 10  |-  ( ph  ->  U. ran  U. ran  (comp `  Y )  e.  U )
4110, 40wunpw 9095 . . . . . . . . 9  |-  ( ph  ->  ~P U. ran  U. ran  (comp `  Y )  e.  U )
4219, 10, 24wunstr 14521 . . . . . . . . 9  |-  ( ph  ->  ( Base `  X
)  e.  U )
4310, 41, 42wunmap 9114 . . . . . . . 8  |-  ( ph  ->  ( ~P U. ran  U.
ran  (comp `  Y )  ^m  ( Base `  X
) )  e.  U
)
4410, 30wunrn 9117 . . . . . . . . . 10  |-  ( ph  ->  ran  ( X Nat  Y
)  e.  U )
4510, 44wununi 9094 . . . . . . . . 9  |-  ( ph  ->  U. ran  ( X Nat 
Y )  e.  U
)
4610, 45, 45wunxp 9112 . . . . . . . 8  |-  ( ph  ->  ( U. ran  ( X Nat  Y )  X.  U. ran  ( X Nat  Y ) )  e.  U )
4710, 43, 46wunpm 9113 . . . . . . 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 5881 . . . . . . . . . . 11  |-  ( 1st `  v )  e.  _V
49 fvex 5881 . . . . . . . . . . . . . 14  |-  ( 2nd `  v )  e.  _V
50 ovex 6319 . . . . . . . . . . . . . . . . 17  |-  ( ~P
U. ran  U. ran  (comp `  Y )  ^m  ( Base `  X ) )  e.  _V
51 ovex 6319 . . . . . . . . . . . . . . . . . . . 20  |-  ( X Nat 
Y )  e.  _V
5251rnex 6728 . . . . . . . . . . . . . . . . . . 19  |-  ran  ( X Nat  Y )  e.  _V
5352uniex 6590 . . . . . . . . . . . . . . . . . 18  |-  U. ran  ( X Nat  Y )  e.  _V
5453, 53xpex 6598 . . . . . . . . . . . . . . . . 17  |-  ( U. ran  ( X Nat  Y )  X.  U. ran  ( X Nat  Y ) )  e. 
_V
55 eqid 2467 . . . . . . . . . . . . . . . . . . . . 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 6320 . . . . . . . . . . . . . . . . . . . . . . . 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 6320 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( <.
( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) )  C_  U. ran  (comp `  Y )
58 rnss 5236 . . . . . . . . . . . . . . . . . . . . . . . . 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 4271 . . . . . . . . . . . . . . . . . . . . . . . . 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 3518 . . . . . . . . . . . . . . . . . . . . . . 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 6319 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( b `  x ) ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  Y )
( ( 1st `  h
) `  x )
) ( a `  x ) )  e. 
_V
6362elpw 4021 . . . . . . . . . . . . . . . . . . . . . . 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 6054 . . . . . . . . . . . . . . . . . . . 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 5881 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  (comp `  Y )  e.  _V
6867rnex 6728 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ran  (comp `  Y )  e.  _V
6968uniex 6590 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  U. ran  (comp `  Y )  e. 
_V
7069rnex 6728 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ran  U. ran  (comp `  Y )  e.  _V
7170uniex 6590 . . . . . . . . . . . . . . . . . . . . . 22  |-  U. ran  U.
ran  (comp `  Y )  e.  _V
7271pwex 4635 . . . . . . . . . . . . . . . . . . . . 21  |-  ~P U. ran  U. ran  (comp `  Y )  e.  _V
73 fvex 5881 . . . . . . . . . . . . . . . . . . . . 21  |-  ( Base `  X )  e.  _V
7472, 73elmap 7457 . . . . . . . . . . . . . . . . . . . 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 2829 . . . . . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . . . . . . 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 6861 . . . . . . . . . . . . . . . . . 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 6320 . . . . . . . . . . . . . . . . . 18  |-  ( g ( X Nat  Y ) h )  C_  U. ran  ( X Nat  Y )
81 ovssunirn 6320 . . . . . . . . . . . . . . . . . 18  |-  ( f ( X Nat  Y ) g )  C_  U. ran  ( X Nat  Y )
82 xpss12 5113 . . . . . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . . . . . . 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 3351 . . . . . . . . . . . . . . 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 3834 . . . . . . . . . . . . . . 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 3351 . . . . . . . . . . . 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 3834 . . . . . . . . . . . 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 2829 . . . . . . . . 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 2467 . . . . . . . . . 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 6861 . . . . . . . . 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 9115 . . . . . 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 9110 . . . . 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 9099 . . . 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 2555 . . 3  |-  ( ph  ->  Q  e.  U )
1031, 13, 16fuccat 15209 . . 3  |-  ( ph  ->  Q  e.  Cat )
104102, 103elind 3693 . 2  |-  ( ph  ->  Q  e.  ( U  i^i  Cat ) )
105104, 11eleqtrrd 2558 1  |-  ( ph  ->  Q  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118   [.wsbc 3336   [_csb 3440    i^i cin 3480    C_ wss 3481   ~Pcpw 4015   {ctp 4036   <.cop 4038   U.cuni 4250    |-> cmpt 4510    X. cxp 5002   ran crn 5005   -->wf 5589   ` cfv 5593  (class class class)co 6294    |-> cmpt2 6296   omcom 6694   1stc1st 6792   2ndc2nd 6793    ^m cmap 7430    ^pm cpm 7431  WUnicwun 9088   1c1 9503   4c4 10597   5c5 10598  ;cdc 10986   ndxcnx 14499   Basecbs 14502   Hom chom 14578  compcco 14579   Catccat 14931    Func cfunc 15093   Nat cnat 15180   FuncCat cfuc 15181  CatCatccatc 15291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-inf2 8068  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-1o 7140  df-oadd 7144  df-omul 7145  df-er 7321  df-ec 7323  df-qs 7327  df-map 7432  df-pm 7433  df-ixp 7480  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-wun 9090  df-ni 9260  df-pli 9261  df-mi 9262  df-lti 9263  df-plpq 9296  df-mpq 9297  df-ltpq 9298  df-enq 9299  df-nq 9300  df-erq 9301  df-plq 9302  df-mq 9303  df-1nq 9304  df-rq 9305  df-ltnq 9306  df-np 9369  df-plp 9371  df-ltp 9373  df-enr 9443  df-nr 9444  df-c 9508  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-nn 10547  df-2 10604  df-3 10605  df-4 10606  df-5 10607  df-6 10608  df-7 10609  df-8 10610  df-9 10611  df-10 10612  df-n0 10806  df-z 10875  df-dec 10987  df-uz 11093  df-fz 11683  df-struct 14504  df-ndx 14505  df-slot 14506  df-base 14507  df-hom 14591  df-cco 14592  df-cat 14935  df-cid 14936  df-func 15097  df-nat 15182  df-fuc 15183  df-catc 15292
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator