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Theorem catcfuccl 14219
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 2404 . . . . 5  |-  ( X 
Func  Y )  =  ( X  Func  Y )
3 eqid 2404 . . . . 5  |-  ( X Nat 
Y )  =  ( X Nat  Y )
4 eqid 2404 . . . . 5  |-  ( Base `  X )  =  (
Base `  X )
5 eqid 2404 . . . . 5  |-  (comp `  Y )  =  (comp `  Y )
6 inss2 3522 . . . . . 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 14207 . . . . . . 7  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
127, 11eleqtrd 2480 . . . . . 6  |-  ( ph  ->  X  e.  ( U  i^i  Cat ) )
136, 12sseldi 3306 . . . . 5  |-  ( ph  ->  X  e.  Cat )
14 catcfuccl.y . . . . . . 7  |-  ( ph  ->  Y  e.  B )
1514, 11eleqtrd 2480 . . . . . 6  |-  ( ph  ->  Y  e.  ( U  i^i  Cat ) )
166, 15sseldi 3306 . . . . 5  |-  ( ph  ->  Y  e.  Cat )
17 eqidd 2405 . . . . 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 14110 . . . 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 13429 . . . . . . 7  |-  Base  = Slot  1
20 catcfuccl.1 . . . . . . . 8  |-  ( ph  ->  om  e.  U )
2110, 20wunndx 13440 . . . . . . 7  |-  ( ph  ->  ndx  e.  U )
2219, 10, 21wunstr 13443 . . . . . 6  |-  ( ph  ->  ( Base `  ndx )  e.  U )
23 inss1 3521 . . . . . . . 8  |-  ( U  i^i  Cat )  C_  U
2423, 12sseldi 3306 . . . . . . 7  |-  ( ph  ->  X  e.  U )
2523, 15sseldi 3306 . . . . . . 7  |-  ( ph  ->  Y  e.  U )
2610, 24, 25wunfunc 14051 . . . . . 6  |-  ( ph  ->  ( X  Func  Y
)  e.  U )
2710, 22, 26wunop 8553 . . . . 5  |-  ( ph  -> 
<. ( Base `  ndx ) ,  ( X  Func  Y ) >.  e.  U
)
28 df-hom 13508 . . . . . . 7  |-  Hom  = Slot ; 1 4
2928, 10, 21wunstr 13443 . . . . . 6  |-  ( ph  ->  (  Hom  `  ndx )  e.  U )
3010, 24, 25wunnat 14108 . . . . . 6  |-  ( ph  ->  ( X Nat  Y )  e.  U )
3110, 29, 30wunop 8553 . . . . 5  |-  ( ph  -> 
<. (  Hom  `  ndx ) ,  ( X Nat  Y ) >.  e.  U
)
32 df-cco 13509 . . . . . . 7  |- comp  = Slot ; 1 5
3332, 10, 21wunstr 13443 . . . . . 6  |-  ( ph  ->  (comp `  ndx )  e.  U )
3410, 26, 26wunxp 8555 . . . . . . . 8  |-  ( ph  ->  ( ( X  Func  Y )  X.  ( X 
Func  Y ) )  e.  U )
3510, 34, 26wunxp 8555 . . . . . . 7  |-  ( ph  ->  ( ( ( X 
Func  Y )  X.  ( X  Func  Y ) )  X.  ( X  Func  Y ) )  e.  U
)
3632, 10, 25wunstr 13443 . . . . . . . . . . . . . 14  |-  ( ph  ->  (comp `  Y )  e.  U )
3710, 36wunrn 8560 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  (comp `  Y
)  e.  U )
3810, 37wununi 8537 . . . . . . . . . . . 12  |-  ( ph  ->  U. ran  (comp `  Y )  e.  U
)
3910, 38wunrn 8560 . . . . . . . . . . 11  |-  ( ph  ->  ran  U. ran  (comp `  Y )  e.  U
)
4010, 39wununi 8537 . . . . . . . . . 10  |-  ( ph  ->  U. ran  U. ran  (comp `  Y )  e.  U )
4110, 40wunpw 8538 . . . . . . . . 9  |-  ( ph  ->  ~P U. ran  U. ran  (comp `  Y )  e.  U )
4219, 10, 24wunstr 13443 . . . . . . . . 9  |-  ( ph  ->  ( Base `  X
)  e.  U )
4310, 41, 42wunmap 8557 . . . . . . . 8  |-  ( ph  ->  ( ~P U. ran  U.
ran  (comp `  Y )  ^m  ( Base `  X
) )  e.  U
)
4410, 30wunrn 8560 . . . . . . . . . 10  |-  ( ph  ->  ran  ( X Nat  Y
)  e.  U )
4510, 44wununi 8537 . . . . . . . . 9  |-  ( ph  ->  U. ran  ( X Nat 
Y )  e.  U
)
4610, 45, 45wunxp 8555 . . . . . . . 8  |-  ( ph  ->  ( U. ran  ( X Nat  Y )  X.  U. ran  ( X Nat  Y ) )  e.  U )
4710, 43, 46wunpm 8556 . . . . . . 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 5701 . . . . . . . . . . 11  |-  ( 1st `  v )  e.  _V
49 fvex 5701 . . . . . . . . . . . . . 14  |-  ( 2nd `  v )  e.  _V
50 ovex 6065 . . . . . . . . . . . . . . . . 17  |-  ( ~P
U. ran  U. ran  (comp `  Y )  ^m  ( Base `  X ) )  e.  _V
51 ovex 6065 . . . . . . . . . . . . . . . . . . . 20  |-  ( X Nat 
Y )  e.  _V
5251rnex 5092 . . . . . . . . . . . . . . . . . . 19  |-  ran  ( X Nat  Y )  e.  _V
5352uniex 4664 . . . . . . . . . . . . . . . . . 18  |-  U. ran  ( X Nat  Y )  e.  _V
5453, 53xpex 4949 . . . . . . . . . . . . . . . . 17  |-  ( U. ran  ( X Nat  Y )  X.  U. ran  ( X Nat  Y ) )  e. 
_V
55 eqid 2404 . . . . . . . . . . . . . . . . . . . . 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 6066 . . . . . . . . . . . . . . . . . . . . . . . 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 6066 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( <.
( ( 1st `  f
) `  x ) ,  ( ( 1st `  g ) `  x
) >. (comp `  Y
) ( ( 1st `  h ) `  x
) )  C_  U. ran  (comp `  Y )
58 rnss 5057 . . . . . . . . . . . . . . . . . . . . . . . . 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 3996 . . . . . . . . . . . . . . . . . . . . . . . . 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 3317 . . . . . . . . . . . . . . . . . . . . . . 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 6065 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( b `  x ) ( <. ( ( 1st `  f ) `  x
) ,  ( ( 1st `  g ) `
 x ) >.
(comp `  Y )
( ( 1st `  h
) `  x )
) ( a `  x ) )  e. 
_V
6362elpw 3765 . . . . . . . . . . . . . . . . . . . . . . 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 201 . . . . . . . . . . . . . . . . . . . . . 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 5851 . . . . . . . . . . . . . . . . . . . 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 5701 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  (comp `  Y )  e.  _V
6867rnex 5092 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ran  (comp `  Y )  e.  _V
6968uniex 4664 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  U. ran  (comp `  Y )  e. 
_V
7069rnex 5092 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ran  U. ran  (comp `  Y )  e.  _V
7170uniex 4664 . . . . . . . . . . . . . . . . . . . . . 22  |-  U. ran  U.
ran  (comp `  Y )  e.  _V
7271pwex 4342 . . . . . . . . . . . . . . . . . . . . 21  |-  ~P U. ran  U. ran  (comp `  Y )  e.  _V
73 fvex 5701 . . . . . . . . . . . . . . . . . . . . 21  |-  ( Base `  X )  e.  _V
7472, 73elmap 7001 . . . . . . . . . . . . . . . . . . . 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 201 . . . . . . . . . . . . . . . . . . 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 2734 . . . . . . . . . . . . . . . . . 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 2404 . . . . . . . . . . . . . . . . . . 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 6377 . . . . . . . . . . . . . . . . . 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 200 . . . . . . . . . . . . . . . . 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 6066 . . . . . . . . . . . . . . . . . 18  |-  ( g ( X Nat  Y ) h )  C_  U. ran  ( X Nat  Y )
81 ovssunirn 6066 . . . . . . . . . . . . . . . . . 18  |-  ( f ( X Nat  Y ) g )  C_  U. ran  ( X Nat  Y )
82 xpss12 4940 . . . . . . . . . . . . . . . . . 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 654 . . . . . . . . . . . . . . . . 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 6993 . . . . . . . . . . . . . . . . 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 655 . . . . . . . . . . . . . . . 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 3135 . . . . . . . . . . . . . . 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 3230 . . . . . . . . . . . . . . 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 202 . . . . . . . . . . . . . 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 3135 . . . . . . . . . . . 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 3230 . . . . . . . . . . . 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 202 . . . . . . . . . . 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 2734 . . . . . . . . 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 2404 . . . . . . . . . 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 6377 . . . . . . . . 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 200 . . . . . . . 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 8558 . . . . . 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 8553 . . . . 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 8542 . . . 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 2478 . . 3  |-  ( ph  ->  Q  e.  U )
1031, 13, 16fuccat 14122 . . 3  |-  ( ph  ->  Q  e.  Cat )
104 elin 3490 . . 3  |-  ( Q  e.  ( U  i^i  Cat )  <->  ( Q  e.  U  /\  Q  e. 
Cat ) )
105102, 103, 104sylanbrc 646 . 2  |-  ( ph  ->  Q  e.  ( U  i^i  Cat ) )
106105, 11eleqtrrd 2481 1  |-  ( ph  ->  Q  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916   [.wsbc 3121   [_csb 3211    i^i cin 3279    C_ wss 3280   ~Pcpw 3759   {ctp 3776   <.cop 3777   U.cuni 3975    e. cmpt 4226   omcom 4804    X. cxp 4835   ran crn 4838   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1stc1st 6306   2ndc2nd 6307    ^m cmap 6977    ^pm cpm 6978  WUnicwun 8531   1c1 8947   4c4 10007   5c5 10008  ;cdc 10338   ndxcnx 13421   Basecbs 13424    Hom chom 13495  compcco 13496   Catccat 13844    Func cfunc 14006   Nat cnat 14093   FuncCat cfuc 14094  CatCatccatc 14204
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-omul 6688  df-er 6864  df-ec 6866  df-qs 6870  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-wun 8533  df-ni 8705  df-pli 8706  df-mi 8707  df-lti 8708  df-plpq 8741  df-mpq 8742  df-ltpq 8743  df-enq 8744  df-nq 8745  df-erq 8746  df-plq 8747  df-mq 8748  df-1nq 8749  df-rq 8750  df-ltnq 8751  df-np 8814  df-plp 8816  df-ltp 8818  df-enr 8890  df-nr 8891  df-c 8952  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-hom 13508  df-cco 13509  df-cat 13848  df-cid 13849  df-func 14010  df-nat 14095  df-fuc 14096  df-catc 14205
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