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Theorem catcxpccl 13981
Description: The category of categories for a weak universe is closed under the product category operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
catcxpccl.c  |-  C  =  (CatCat `  U )
catcxpccl.b  |-  B  =  ( Base `  C
)
catcxpccl.o  |-  T  =  ( X  X.c  Y )
catcxpccl.u  |-  ( ph  ->  U  e. WUni )
catcxpccl.1  |-  ( ph  ->  om  e.  U )
catcxpccl.x  |-  ( ph  ->  X  e.  B )
catcxpccl.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
catcxpccl  |-  ( ph  ->  T  e.  B )

Proof of Theorem catcxpccl
Dummy variables  f 
g  u  v  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catcxpccl.o . . . . 5  |-  T  =  ( X  X.c  Y )
2 eqid 2283 . . . . 5  |-  ( Base `  X )  =  (
Base `  X )
3 eqid 2283 . . . . 5  |-  ( Base `  Y )  =  (
Base `  Y )
4 eqid 2283 . . . . 5  |-  (  Hom  `  X )  =  (  Hom  `  X )
5 eqid 2283 . . . . 5  |-  (  Hom  `  Y )  =  (  Hom  `  Y )
6 eqid 2283 . . . . 5  |-  (comp `  X )  =  (comp `  X )
7 eqid 2283 . . . . 5  |-  (comp `  Y )  =  (comp `  Y )
8 catcxpccl.x . . . . 5  |-  ( ph  ->  X  e.  B )
9 catcxpccl.y . . . . 5  |-  ( ph  ->  Y  e.  B )
10 eqidd 2284 . . . . 5  |-  ( ph  ->  ( ( Base `  X
)  X.  ( Base `  Y ) )  =  ( ( Base `  X
)  X.  ( Base `  Y ) ) )
111, 2, 3xpcbas 13952 . . . . . . 7  |-  ( (
Base `  X )  X.  ( Base `  Y
) )  =  (
Base `  T )
12 eqid 2283 . . . . . . 7  |-  (  Hom  `  T )  =  (  Hom  `  T )
131, 11, 4, 5, 12xpchomfval 13953 . . . . . 6  |-  (  Hom  `  T )  =  ( u  e.  ( (
Base `  X )  X.  ( Base `  Y
) ) ,  v  e.  ( ( Base `  X )  X.  ( Base `  Y ) ) 
|->  ( ( ( 1st `  u ) (  Hom  `  X ) ( 1st `  v ) )  X.  ( ( 2nd `  u
) (  Hom  `  Y
) ( 2nd `  v
) ) ) )
1413a1i 10 . . . . 5  |-  ( ph  ->  (  Hom  `  T
)  =  ( u  e.  ( ( Base `  X )  X.  ( Base `  Y ) ) ,  v  e.  ( ( Base `  X
)  X.  ( Base `  Y ) )  |->  ( ( ( 1st `  u
) (  Hom  `  X
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  Y
) ( 2nd `  v
) ) ) ) )
15 eqidd 2284 . . . . 5  |-  ( ph  ->  ( x  e.  ( ( ( Base `  X
)  X.  ( Base `  Y ) )  X.  ( ( Base `  X
)  X.  ( Base `  Y ) ) ) ,  y  e.  ( ( Base `  X
)  X.  ( Base `  Y ) )  |->  ( g  e.  ( ( 2nd `  x ) (  Hom  `  T
) y ) ,  f  e.  ( (  Hom  `  T ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )  =  ( x  e.  ( ( ( Base `  X
)  X.  ( Base `  Y ) )  X.  ( ( Base `  X
)  X.  ( Base `  Y ) ) ) ,  y  e.  ( ( Base `  X
)  X.  ( Base `  Y ) )  |->  ( g  e.  ( ( 2nd `  x ) (  Hom  `  T
) y ) ,  f  e.  ( (  Hom  `  T ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) )
161, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15xpcval 13951 . . . 4  |-  ( ph  ->  T  =  { <. (
Base `  ndx ) ,  ( ( Base `  X
)  X.  ( Base `  Y ) ) >. ,  <. (  Hom  `  ndx ) ,  (  Hom  `  T ) >. ,  <. (comp `  ndx ) ,  ( x  e.  ( ( ( Base `  X
)  X.  ( Base `  Y ) )  X.  ( ( Base `  X
)  X.  ( Base `  Y ) ) ) ,  y  e.  ( ( Base `  X
)  X.  ( Base `  Y ) )  |->  ( g  e.  ( ( 2nd `  x ) (  Hom  `  T
) y ) ,  f  e.  ( (  Hom  `  T ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >. } )
17 catcxpccl.u . . . . 5  |-  ( ph  ->  U  e. WUni )
18 df-base 13153 . . . . . . 7  |-  Base  = Slot  1
19 catcxpccl.1 . . . . . . . 8  |-  ( ph  ->  om  e.  U )
2017, 19wunndx 13164 . . . . . . 7  |-  ( ph  ->  ndx  e.  U )
2118, 17, 20wunstr 13167 . . . . . 6  |-  ( ph  ->  ( Base `  ndx )  e.  U )
22 inss1 3389 . . . . . . . . 9  |-  ( U  i^i  Cat )  C_  U
23 catcxpccl.c . . . . . . . . . . 11  |-  C  =  (CatCat `  U )
24 catcxpccl.b . . . . . . . . . . 11  |-  B  =  ( Base `  C
)
2523, 24, 17catcbas 13929 . . . . . . . . . 10  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
268, 25eleqtrd 2359 . . . . . . . . 9  |-  ( ph  ->  X  e.  ( U  i^i  Cat ) )
2722, 26sseldi 3178 . . . . . . . 8  |-  ( ph  ->  X  e.  U )
2818, 17, 27wunstr 13167 . . . . . . 7  |-  ( ph  ->  ( Base `  X
)  e.  U )
299, 25eleqtrd 2359 . . . . . . . . 9  |-  ( ph  ->  Y  e.  ( U  i^i  Cat ) )
3022, 29sseldi 3178 . . . . . . . 8  |-  ( ph  ->  Y  e.  U )
3118, 17, 30wunstr 13167 . . . . . . 7  |-  ( ph  ->  ( Base `  Y
)  e.  U )
3217, 28, 31wunxp 8346 . . . . . 6  |-  ( ph  ->  ( ( Base `  X
)  X.  ( Base `  Y ) )  e.  U )
3317, 21, 32wunop 8344 . . . . 5  |-  ( ph  -> 
<. ( Base `  ndx ) ,  ( ( Base `  X )  X.  ( Base `  Y
) ) >.  e.  U
)
34 df-hom 13232 . . . . . . 7  |-  Hom  = Slot ; 1 4
3534, 17, 20wunstr 13167 . . . . . 6  |-  ( ph  ->  (  Hom  `  ndx )  e.  U )
3617, 32, 32wunxp 8346 . . . . . . . 8  |-  ( ph  ->  ( ( ( Base `  X )  X.  ( Base `  Y ) )  X.  ( ( Base `  X )  X.  ( Base `  Y ) ) )  e.  U )
3734, 17, 27wunstr 13167 . . . . . . . . . . . 12  |-  ( ph  ->  (  Hom  `  X
)  e.  U )
3817, 37wunrn 8351 . . . . . . . . . . 11  |-  ( ph  ->  ran  (  Hom  `  X
)  e.  U )
3917, 38wununi 8328 . . . . . . . . . 10  |-  ( ph  ->  U. ran  (  Hom  `  X )  e.  U
)
4034, 17, 30wunstr 13167 . . . . . . . . . . . 12  |-  ( ph  ->  (  Hom  `  Y
)  e.  U )
4117, 40wunrn 8351 . . . . . . . . . . 11  |-  ( ph  ->  ran  (  Hom  `  Y
)  e.  U )
4217, 41wununi 8328 . . . . . . . . . 10  |-  ( ph  ->  U. ran  (  Hom  `  Y )  e.  U
)
4317, 39, 42wunxp 8346 . . . . . . . . 9  |-  ( ph  ->  ( U. ran  (  Hom  `  X )  X. 
U. ran  (  Hom  `  Y ) )  e.  U )
4417, 43wunpw 8329 . . . . . . . 8  |-  ( ph  ->  ~P ( U. ran  (  Hom  `  X )  X.  U. ran  (  Hom  `  Y ) )  e.  U )
45 ovssunirn 5884 . . . . . . . . . . . . 13  |-  ( ( 1st `  u ) (  Hom  `  X
) ( 1st `  v
) )  C_  U. ran  (  Hom  `  X )
46 ovssunirn 5884 . . . . . . . . . . . . 13  |-  ( ( 2nd `  u ) (  Hom  `  Y
) ( 2nd `  v
) )  C_  U. ran  (  Hom  `  Y )
47 xpss12 4792 . . . . . . . . . . . . 13  |-  ( ( ( ( 1st `  u
) (  Hom  `  X
) ( 1st `  v
) )  C_  U. ran  (  Hom  `  X )  /\  ( ( 2nd `  u
) (  Hom  `  Y
) ( 2nd `  v
) )  C_  U. ran  (  Hom  `  Y )
)  ->  ( (
( 1st `  u
) (  Hom  `  X
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  Y
) ( 2nd `  v
) ) )  C_  ( U. ran  (  Hom  `  X )  X.  U. ran  (  Hom  `  Y
) ) )
4845, 46, 47mp2an 653 . . . . . . . . . . . 12  |-  ( ( ( 1st `  u
) (  Hom  `  X
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  Y
) ( 2nd `  v
) ) )  C_  ( U. ran  (  Hom  `  X )  X.  U. ran  (  Hom  `  Y
) )
49 ovex 5883 . . . . . . . . . . . . . 14  |-  ( ( 1st `  u ) (  Hom  `  X
) ( 1st `  v
) )  e.  _V
50 ovex 5883 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  u ) (  Hom  `  Y
) ( 2nd `  v
) )  e.  _V
5149, 50xpex 4801 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  u
) (  Hom  `  X
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  Y
) ( 2nd `  v
) ) )  e. 
_V
5251elpw 3631 . . . . . . . . . . . 12  |-  ( ( ( ( 1st `  u
) (  Hom  `  X
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  Y
) ( 2nd `  v
) ) )  e. 
~P ( U. ran  (  Hom  `  X )  X.  U. ran  (  Hom  `  Y ) )  <->  ( (
( 1st `  u
) (  Hom  `  X
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  Y
) ( 2nd `  v
) ) )  C_  ( U. ran  (  Hom  `  X )  X.  U. ran  (  Hom  `  Y
) ) )
5348, 52mpbir 200 . . . . . . . . . . 11  |-  ( ( ( 1st `  u
) (  Hom  `  X
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  Y
) ( 2nd `  v
) ) )  e. 
~P ( U. ran  (  Hom  `  X )  X.  U. ran  (  Hom  `  Y ) )
5453rgen2w 2611 . . . . . . . . . 10  |-  A. u  e.  ( ( Base `  X
)  X.  ( Base `  Y ) ) A. v  e.  ( ( Base `  X )  X.  ( Base `  Y
) ) ( ( ( 1st `  u
) (  Hom  `  X
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  Y
) ( 2nd `  v
) ) )  e. 
~P ( U. ran  (  Hom  `  X )  X.  U. ran  (  Hom  `  Y ) )
55 eqid 2283 . . . . . . . . . . 11  |-  ( u  e.  ( ( Base `  X )  X.  ( Base `  Y ) ) ,  v  e.  ( ( Base `  X
)  X.  ( Base `  Y ) )  |->  ( ( ( 1st `  u
) (  Hom  `  X
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  Y
) ( 2nd `  v
) ) ) )  =  ( u  e.  ( ( Base `  X
)  X.  ( Base `  Y ) ) ,  v  e.  ( (
Base `  X )  X.  ( Base `  Y
) )  |->  ( ( ( 1st `  u
) (  Hom  `  X
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  Y
) ( 2nd `  v
) ) ) )
5655fmpt2 6191 . . . . . . . . . 10  |-  ( A. u  e.  ( ( Base `  X )  X.  ( Base `  Y
) ) A. v  e.  ( ( Base `  X
)  X.  ( Base `  Y ) ) ( ( ( 1st `  u
) (  Hom  `  X
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  Y
) ( 2nd `  v
) ) )  e. 
~P ( U. ran  (  Hom  `  X )  X.  U. ran  (  Hom  `  Y ) )  <->  ( u  e.  ( ( Base `  X
)  X.  ( Base `  Y ) ) ,  v  e.  ( (
Base `  X )  X.  ( Base `  Y
) )  |->  ( ( ( 1st `  u
) (  Hom  `  X
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  Y
) ( 2nd `  v
) ) ) ) : ( ( (
Base `  X )  X.  ( Base `  Y
) )  X.  (
( Base `  X )  X.  ( Base `  Y
) ) ) --> ~P ( U. ran  (  Hom  `  X )  X. 
U. ran  (  Hom  `  Y ) ) )
5754, 56mpbi 199 . . . . . . . . 9  |-  ( u  e.  ( ( Base `  X )  X.  ( Base `  Y ) ) ,  v  e.  ( ( Base `  X
)  X.  ( Base `  Y ) )  |->  ( ( ( 1st `  u
) (  Hom  `  X
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  Y
) ( 2nd `  v
) ) ) ) : ( ( (
Base `  X )  X.  ( Base `  Y
) )  X.  (
( Base `  X )  X.  ( Base `  Y
) ) ) --> ~P ( U. ran  (  Hom  `  X )  X. 
U. ran  (  Hom  `  Y ) )
5857a1i 10 . . . . . . . 8  |-  ( ph  ->  ( u  e.  ( ( Base `  X
)  X.  ( Base `  Y ) ) ,  v  e.  ( (
Base `  X )  X.  ( Base `  Y
) )  |->  ( ( ( 1st `  u
) (  Hom  `  X
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  Y
) ( 2nd `  v
) ) ) ) : ( ( (
Base `  X )  X.  ( Base `  Y
) )  X.  (
( Base `  X )  X.  ( Base `  Y
) ) ) --> ~P ( U. ran  (  Hom  `  X )  X. 
U. ran  (  Hom  `  Y ) ) )
5917, 36, 44, 58wunf 8349 . . . . . . 7  |-  ( ph  ->  ( u  e.  ( ( Base `  X
)  X.  ( Base `  Y ) ) ,  v  e.  ( (
Base `  X )  X.  ( Base `  Y
) )  |->  ( ( ( 1st `  u
) (  Hom  `  X
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  Y
) ( 2nd `  v
) ) ) )  e.  U )
6013, 59syl5eqel 2367 . . . . . 6  |-  ( ph  ->  (  Hom  `  T
)  e.  U )
6117, 35, 60wunop 8344 . . . . 5  |-  ( ph  -> 
<. (  Hom  `  ndx ) ,  (  Hom  `  T ) >.  e.  U
)
62 df-cco 13233 . . . . . . 7  |- comp  = Slot ; 1 5
6362, 17, 20wunstr 13167 . . . . . 6  |-  ( ph  ->  (comp `  ndx )  e.  U )
6417, 36, 32wunxp 8346 . . . . . . 7  |-  ( ph  ->  ( ( ( (
Base `  X )  X.  ( Base `  Y
) )  X.  (
( Base `  X )  X.  ( Base `  Y
) ) )  X.  ( ( Base `  X
)  X.  ( Base `  Y ) ) )  e.  U )
6562, 17, 27wunstr 13167 . . . . . . . . . . . . . 14  |-  ( ph  ->  (comp `  X )  e.  U )
6617, 65wunrn 8351 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  (comp `  X
)  e.  U )
6717, 66wununi 8328 . . . . . . . . . . . 12  |-  ( ph  ->  U. ran  (comp `  X )  e.  U
)
6817, 67wunrn 8351 . . . . . . . . . . 11  |-  ( ph  ->  ran  U. ran  (comp `  X )  e.  U
)
6917, 68wununi 8328 . . . . . . . . . 10  |-  ( ph  ->  U. ran  U. ran  (comp `  X )  e.  U )
7017, 69wunpw 8329 . . . . . . . . 9  |-  ( ph  ->  ~P U. ran  U. ran  (comp `  X )  e.  U )
7162, 17, 30wunstr 13167 . . . . . . . . . . . . . 14  |-  ( ph  ->  (comp `  Y )  e.  U )
7217, 71wunrn 8351 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  (comp `  Y
)  e.  U )
7317, 72wununi 8328 . . . . . . . . . . . 12  |-  ( ph  ->  U. ran  (comp `  Y )  e.  U
)
7417, 73wunrn 8351 . . . . . . . . . . 11  |-  ( ph  ->  ran  U. ran  (comp `  Y )  e.  U
)
7517, 74wununi 8328 . . . . . . . . . 10  |-  ( ph  ->  U. ran  U. ran  (comp `  Y )  e.  U )
7617, 75wunpw 8329 . . . . . . . . 9  |-  ( ph  ->  ~P U. ran  U. ran  (comp `  Y )  e.  U )
7717, 70, 76wunxp 8346 . . . . . . . 8  |-  ( ph  ->  ( ~P U. ran  U.
ran  (comp `  X )  X.  ~P U. ran  U. ran  (comp `  Y )
)  e.  U )
7817, 60wunrn 8351 . . . . . . . . . 10  |-  ( ph  ->  ran  (  Hom  `  T
)  e.  U )
7917, 78wununi 8328 . . . . . . . . 9  |-  ( ph  ->  U. ran  (  Hom  `  T )  e.  U
)
8017, 79, 79wunxp 8346 . . . . . . . 8  |-  ( ph  ->  ( U. ran  (  Hom  `  T )  X. 
U. ran  (  Hom  `  T ) )  e.  U )
8117, 77, 80wunpm 8347 . . . . . . 7  |-  ( ph  ->  ( ( ~P U. ran  U. ran  (comp `  X )  X.  ~P U.
ran  U. ran  (comp `  Y ) )  ^pm  ( U. ran  (  Hom  `  T )  X.  U. ran  (  Hom  `  T
) ) )  e.  U )
82 fvex 5539 . . . . . . . . . . . . . . . . 17  |-  (comp `  X )  e.  _V
8382rnex 4942 . . . . . . . . . . . . . . . 16  |-  ran  (comp `  X )  e.  _V
8483uniex 4516 . . . . . . . . . . . . . . 15  |-  U. ran  (comp `  X )  e. 
_V
8584rnex 4942 . . . . . . . . . . . . . 14  |-  ran  U. ran  (comp `  X )  e.  _V
8685uniex 4516 . . . . . . . . . . . . 13  |-  U. ran  U.
ran  (comp `  X )  e.  _V
8786pwex 4193 . . . . . . . . . . . 12  |-  ~P U. ran  U. ran  (comp `  X )  e.  _V
88 fvex 5539 . . . . . . . . . . . . . . . . 17  |-  (comp `  Y )  e.  _V
8988rnex 4942 . . . . . . . . . . . . . . . 16  |-  ran  (comp `  Y )  e.  _V
9089uniex 4516 . . . . . . . . . . . . . . 15  |-  U. ran  (comp `  Y )  e. 
_V
9190rnex 4942 . . . . . . . . . . . . . 14  |-  ran  U. ran  (comp `  Y )  e.  _V
9291uniex 4516 . . . . . . . . . . . . 13  |-  U. ran  U.
ran  (comp `  Y )  e.  _V
9392pwex 4193 . . . . . . . . . . . 12  |-  ~P U. ran  U. ran  (comp `  Y )  e.  _V
9487, 93xpex 4801 . . . . . . . . . . 11  |-  ( ~P
U. ran  U. ran  (comp `  X )  X.  ~P U.
ran  U. ran  (comp `  Y ) )  e. 
_V
95 fvex 5539 . . . . . . . . . . . . . 14  |-  (  Hom  `  T )  e.  _V
9695rnex 4942 . . . . . . . . . . . . 13  |-  ran  (  Hom  `  T )  e. 
_V
9796uniex 4516 . . . . . . . . . . . 12  |-  U. ran  (  Hom  `  T )  e.  _V
9897, 97xpex 4801 . . . . . . . . . . 11  |-  ( U. ran  (  Hom  `  T
)  X.  U. ran  (  Hom  `  T )
)  e.  _V
99 ovssunirn 5884 . . . . . . . . . . . . . . . 16  |-  ( ( 1st `  g ) ( <. ( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  X )
( 1st `  y
) ) ( 1st `  f ) )  C_  U.
ran  ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) )
100 ovssunirn 5884 . . . . . . . . . . . . . . . . 17  |-  ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  X )
( 1st `  y
) )  C_  U. ran  (comp `  X )
101 rnss 4907 . . . . . . . . . . . . . . . . 17  |-  ( (
<. ( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  X )
( 1st `  y
) )  C_  U. ran  (comp `  X )  ->  ran  ( <. ( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  X )
( 1st `  y
) )  C_  ran  U.
ran  (comp `  X )
)
102 uniss 3848 . . . . . . . . . . . . . . . . 17  |-  ( ran  ( <. ( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  X )
( 1st `  y
) )  C_  ran  U.
ran  (comp `  X )  ->  U. ran  ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  X )
( 1st `  y
) )  C_  U. ran  U.
ran  (comp `  X )
)
103100, 101, 102mp2b 9 . . . . . . . . . . . . . . . 16  |-  U. ran  ( <. ( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  X )
( 1st `  y
) )  C_  U. ran  U.
ran  (comp `  X )
10499, 103sstri 3188 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  g ) ( <. ( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  X )
( 1st `  y
) ) ( 1st `  f ) )  C_  U.
ran  U. ran  (comp `  X )
105 ovex 5883 . . . . . . . . . . . . . . . 16  |-  ( ( 1st `  g ) ( <. ( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  X )
( 1st `  y
) ) ( 1st `  f ) )  e. 
_V
106105elpw 3631 . . . . . . . . . . . . . . 15  |-  ( ( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) ) ( 1st `  f ) )  e. 
~P U. ran  U. ran  (comp `  X )  <->  ( ( 1st `  g ) (
<. ( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  X )
( 1st `  y
) ) ( 1st `  f ) )  C_  U.
ran  U. ran  (comp `  X ) )
107104, 106mpbir 200 . . . . . . . . . . . . . 14  |-  ( ( 1st `  g ) ( <. ( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  X )
( 1st `  y
) ) ( 1st `  f ) )  e. 
~P U. ran  U. ran  (comp `  X )
108 ovssunirn 5884 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  g ) ( <. ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>. (comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) )  C_  U.
ran  ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) )
109 ovssunirn 5884 . . . . . . . . . . . . . . . . 17  |-  ( <.
( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>. (comp `  Y )
( 2nd `  y
) )  C_  U. ran  (comp `  Y )
110 rnss 4907 . . . . . . . . . . . . . . . . 17  |-  ( (
<. ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>. (comp `  Y )
( 2nd `  y
) )  C_  U. ran  (comp `  Y )  ->  ran  ( <. ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>. (comp `  Y )
( 2nd `  y
) )  C_  ran  U.
ran  (comp `  Y )
)
111 uniss 3848 . . . . . . . . . . . . . . . . 17  |-  ( ran  ( <. ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>. (comp `  Y )
( 2nd `  y
) )  C_  ran  U.
ran  (comp `  Y )  ->  U. ran  ( <.
( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>. (comp `  Y )
( 2nd `  y
) )  C_  U. ran  U.
ran  (comp `  Y )
)
112109, 110, 111mp2b 9 . . . . . . . . . . . . . . . 16  |-  U. ran  ( <. ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>. (comp `  Y )
( 2nd `  y
) )  C_  U. ran  U.
ran  (comp `  Y )
113108, 112sstri 3188 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  g ) ( <. ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>. (comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) )  C_  U.
ran  U. ran  (comp `  Y )
114 ovex 5883 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  g ) ( <. ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>. (comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) )  e. 
_V
115114elpw 3631 . . . . . . . . . . . . . . 15  |-  ( ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) )  e. 
~P U. ran  U. ran  (comp `  Y )  <->  ( ( 2nd `  g ) (
<. ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>. (comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) )  C_  U.
ran  U. ran  (comp `  Y ) )
116113, 115mpbir 200 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  g ) ( <. ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>. (comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) )  e. 
~P U. ran  U. ran  (comp `  Y )
117 opelxpi 4721 . . . . . . . . . . . . . 14  |-  ( ( ( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) ) ( 1st `  f ) )  e. 
~P U. ran  U. ran  (comp `  X )  /\  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) )  e. 
~P U. ran  U. ran  (comp `  Y ) )  ->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  X )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) ) >.  e.  ( ~P U. ran  U.
ran  (comp `  X )  X.  ~P U. ran  U. ran  (comp `  Y )
) )
118107, 116, 117mp2an 653 . . . . . . . . . . . . 13  |-  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) ) >.  e.  ( ~P U. ran  U.
ran  (comp `  X )  X.  ~P U. ran  U. ran  (comp `  Y )
)
119118rgen2w 2611 . . . . . . . . . . . 12  |-  A. g  e.  ( ( 2nd `  x
) (  Hom  `  T
) y ) A. f  e.  ( (  Hom  `  T ) `  x ) <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) ) >.  e.  ( ~P U. ran  U.
ran  (comp `  X )  X.  ~P U. ran  U. ran  (comp `  Y )
)
120 eqid 2283 . . . . . . . . . . . . 13  |-  ( g  e.  ( ( 2nd `  x ) (  Hom  `  T ) y ) ,  f  e.  ( (  Hom  `  T
) `  x )  |-> 
<. ( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
)  =  ( g  e.  ( ( 2nd `  x ) (  Hom  `  T ) y ) ,  f  e.  ( (  Hom  `  T
) `  x )  |-> 
<. ( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
)
121120fmpt2 6191 . . . . . . . . . . . 12  |-  ( A. g  e.  ( ( 2nd `  x ) (  Hom  `  T )
y ) A. f  e.  ( (  Hom  `  T
) `  x ) <. ( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) ) >.  e.  ( ~P U. ran  U.
ran  (comp `  X )  X.  ~P U. ran  U. ran  (comp `  Y )
)  <->  ( g  e.  ( ( 2nd `  x
) (  Hom  `  T
) y ) ,  f  e.  ( (  Hom  `  T ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) : ( ( ( 2nd `  x
) (  Hom  `  T
) y )  X.  ( (  Hom  `  T
) `  x )
) --> ( ~P U. ran  U. ran  (comp `  X )  X.  ~P U.
ran  U. ran  (comp `  Y ) ) )
122119, 121mpbi 199 . . . . . . . . . . 11  |-  ( g  e.  ( ( 2nd `  x ) (  Hom  `  T ) y ) ,  f  e.  ( (  Hom  `  T
) `  x )  |-> 
<. ( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) : ( ( ( 2nd `  x
) (  Hom  `  T
) y )  X.  ( (  Hom  `  T
) `  x )
) --> ( ~P U. ran  U. ran  (comp `  X )  X.  ~P U.
ran  U. ran  (comp `  Y ) )
123 ovssunirn 5884 . . . . . . . . . . . 12  |-  ( ( 2nd `  x ) (  Hom  `  T
) y )  C_  U.
ran  (  Hom  `  T
)
124 fvssunirn 5551 . . . . . . . . . . . 12  |-  ( (  Hom  `  T ) `  x )  C_  U. ran  (  Hom  `  T )
125 xpss12 4792 . . . . . . . . . . . 12  |-  ( ( ( ( 2nd `  x
) (  Hom  `  T
) y )  C_  U.
ran  (  Hom  `  T
)  /\  ( (  Hom  `  T ) `  x )  C_  U. ran  (  Hom  `  T )
)  ->  ( (
( 2nd `  x
) (  Hom  `  T
) y )  X.  ( (  Hom  `  T
) `  x )
)  C_  ( U. ran  (  Hom  `  T
)  X.  U. ran  (  Hom  `  T )
) )
126123, 124, 125mp2an 653 . . . . . . . . . . 11  |-  ( ( ( 2nd `  x
) (  Hom  `  T
) y )  X.  ( (  Hom  `  T
) `  x )
)  C_  ( U. ran  (  Hom  `  T
)  X.  U. ran  (  Hom  `  T )
)
127 elpm2r 6788 . . . . . . . . . . 11  |-  ( ( ( ( ~P U. ran  U. ran  (comp `  X )  X.  ~P U.
ran  U. ran  (comp `  Y ) )  e. 
_V  /\  ( U. ran  (  Hom  `  T
)  X.  U. ran  (  Hom  `  T )
)  e.  _V )  /\  ( ( g  e.  ( ( 2nd `  x
) (  Hom  `  T
) y ) ,  f  e.  ( (  Hom  `  T ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) : ( ( ( 2nd `  x
) (  Hom  `  T
) y )  X.  ( (  Hom  `  T
) `  x )
) --> ( ~P U. ran  U. ran  (comp `  X )  X.  ~P U.
ran  U. ran  (comp `  Y ) )  /\  ( ( ( 2nd `  x ) (  Hom  `  T ) y )  X.  ( (  Hom  `  T ) `  x
) )  C_  ( U. ran  (  Hom  `  T
)  X.  U. ran  (  Hom  `  T )
) ) )  -> 
( g  e.  ( ( 2nd `  x
) (  Hom  `  T
) y ) ,  f  e.  ( (  Hom  `  T ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
)  e.  ( ( ~P U. ran  U. ran  (comp `  X )  X.  ~P U. ran  U. ran  (comp `  Y )
)  ^pm  ( U. ran  (  Hom  `  T
)  X.  U. ran  (  Hom  `  T )
) ) )
12894, 98, 122, 126, 127mp4an 654 . . . . . . . . . 10  |-  ( g  e.  ( ( 2nd `  x ) (  Hom  `  T ) y ) ,  f  e.  ( (  Hom  `  T
) `  x )  |-> 
<. ( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
)  e.  ( ( ~P U. ran  U. ran  (comp `  X )  X.  ~P U. ran  U. ran  (comp `  Y )
)  ^pm  ( U. ran  (  Hom  `  T
)  X.  U. ran  (  Hom  `  T )
) )
129128rgen2w 2611 . . . . . . . . 9  |-  A. x  e.  ( ( ( Base `  X )  X.  ( Base `  Y ) )  X.  ( ( Base `  X )  X.  ( Base `  Y ) ) ) A. y  e.  ( ( Base `  X
)  X.  ( Base `  Y ) ) ( g  e.  ( ( 2nd `  x ) (  Hom  `  T
) y ) ,  f  e.  ( (  Hom  `  T ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
)  e.  ( ( ~P U. ran  U. ran  (comp `  X )  X.  ~P U. ran  U. ran  (comp `  Y )
)  ^pm  ( U. ran  (  Hom  `  T
)  X.  U. ran  (  Hom  `  T )
) )
130 eqid 2283 . . . . . . . . . 10  |-  ( x  e.  ( ( (
Base `  X )  X.  ( Base `  Y
) )  X.  (
( Base `  X )  X.  ( Base `  Y
) ) ) ,  y  e.  ( (
Base `  X )  X.  ( Base `  Y
) )  |->  ( g  e.  ( ( 2nd `  x ) (  Hom  `  T ) y ) ,  f  e.  ( (  Hom  `  T
) `  x )  |-> 
<. ( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )  =  ( x  e.  ( ( ( Base `  X
)  X.  ( Base `  Y ) )  X.  ( ( Base `  X
)  X.  ( Base `  Y ) ) ) ,  y  e.  ( ( Base `  X
)  X.  ( Base `  Y ) )  |->  ( g  e.  ( ( 2nd `  x ) (  Hom  `  T
) y ) ,  f  e.  ( (  Hom  `  T ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )
131130fmpt2 6191 . . . . . . . . 9  |-  ( A. x  e.  ( (
( Base `  X )  X.  ( Base `  Y
) )  X.  (
( Base `  X )  X.  ( Base `  Y
) ) ) A. y  e.  ( ( Base `  X )  X.  ( Base `  Y
) ) ( g  e.  ( ( 2nd `  x ) (  Hom  `  T ) y ) ,  f  e.  ( (  Hom  `  T
) `  x )  |-> 
<. ( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
)  e.  ( ( ~P U. ran  U. ran  (comp `  X )  X.  ~P U. ran  U. ran  (comp `  Y )
)  ^pm  ( U. ran  (  Hom  `  T
)  X.  U. ran  (  Hom  `  T )
) )  <->  ( x  e.  ( ( ( Base `  X )  X.  ( Base `  Y ) )  X.  ( ( Base `  X )  X.  ( Base `  Y ) ) ) ,  y  e.  ( ( Base `  X
)  X.  ( Base `  Y ) )  |->  ( g  e.  ( ( 2nd `  x ) (  Hom  `  T
) y ) ,  f  e.  ( (  Hom  `  T ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) : ( ( ( ( Base `  X )  X.  ( Base `  Y ) )  X.  ( ( Base `  X )  X.  ( Base `  Y ) ) )  X.  ( (
Base `  X )  X.  ( Base `  Y
) ) ) --> ( ( ~P U. ran  U.
ran  (comp `  X )  X.  ~P U. ran  U. ran  (comp `  Y )
)  ^pm  ( U. ran  (  Hom  `  T
)  X.  U. ran  (  Hom  `  T )
) ) )
132129, 131mpbi 199 . . . . . . . 8  |-  ( x  e.  ( ( (
Base `  X )  X.  ( Base `  Y
) )  X.  (
( Base `  X )  X.  ( Base `  Y
) ) ) ,  y  e.  ( (
Base `  X )  X.  ( Base `  Y
) )  |->  ( g  e.  ( ( 2nd `  x ) (  Hom  `  T ) y ) ,  f  e.  ( (  Hom  `  T
) `  x )  |-> 
<. ( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) : ( ( ( ( Base `  X )  X.  ( Base `  Y ) )  X.  ( ( Base `  X )  X.  ( Base `  Y ) ) )  X.  ( (
Base `  X )  X.  ( Base `  Y
) ) ) --> ( ( ~P U. ran  U.
ran  (comp `  X )  X.  ~P U. ran  U. ran  (comp `  Y )
)  ^pm  ( U. ran  (  Hom  `  T
)  X.  U. ran  (  Hom  `  T )
) )
133132a1i 10 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( ( ( Base `  X
)  X.  ( Base `  Y ) )  X.  ( ( Base `  X
)  X.  ( Base `  Y ) ) ) ,  y  e.  ( ( Base `  X
)  X.  ( Base `  Y ) )  |->  ( g  e.  ( ( 2nd `  x ) (  Hom  `  T
) y ) ,  f  e.  ( (  Hom  `  T ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) : ( ( ( ( Base `  X )  X.  ( Base `  Y ) )  X.  ( ( Base `  X )  X.  ( Base `  Y ) ) )  X.  ( (
Base `  X )  X.  ( Base `  Y
) ) ) --> ( ( ~P U. ran  U.
ran  (comp `  X )  X.  ~P U. ran  U. ran  (comp `  Y )
)  ^pm  ( U. ran  (  Hom  `  T
)  X.  U. ran  (  Hom  `  T )
) ) )
13417, 64, 81, 133wunf 8349 . . . . . 6  |-  ( ph  ->  ( x  e.  ( ( ( Base `  X
)  X.  ( Base `  Y ) )  X.  ( ( Base `  X
)  X.  ( Base `  Y ) ) ) ,  y  e.  ( ( Base `  X
)  X.  ( Base `  Y ) )  |->  ( g  e.  ( ( 2nd `  x ) (  Hom  `  T
) y ) ,  f  e.  ( (  Hom  `  T ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )  e.  U
)
13517, 63, 134wunop 8344 . . . . 5  |-  ( ph  -> 
<. (comp `  ndx ) ,  ( x  e.  ( ( ( Base `  X
)  X.  ( Base `  Y ) )  X.  ( ( Base `  X
)  X.  ( Base `  Y ) ) ) ,  y  e.  ( ( Base `  X
)  X.  ( Base `  Y ) )  |->  ( g  e.  ( ( 2nd `  x ) (  Hom  `  T
) y ) ,  f  e.  ( (  Hom  `  T ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >.  e.  U
)
13617, 33, 61, 135wuntp 8333 . . . 4  |-  ( ph  ->  { <. ( Base `  ndx ) ,  ( ( Base `  X )  X.  ( Base `  Y
) ) >. ,  <. (  Hom  `  ndx ) ,  (  Hom  `  T
) >. ,  <. (comp ` 
ndx ) ,  ( x  e.  ( ( ( Base `  X
)  X.  ( Base `  Y ) )  X.  ( ( Base `  X
)  X.  ( Base `  Y ) ) ) ,  y  e.  ( ( Base `  X
)  X.  ( Base `  Y ) )  |->  ( g  e.  ( ( 2nd `  x ) (  Hom  `  T
) y ) ,  f  e.  ( (  Hom  `  T ) `  x )  |->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.
(comp `  X )
( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.
(comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) ) >.
) ) >. }  e.  U )
13716, 136eqeltrd 2357 . . 3  |-  ( ph  ->  T  e.  U )
138 inss2 3390 . . . . 5  |-  ( U  i^i  Cat )  C_  Cat
139138, 26sseldi 3178 . . . 4  |-  ( ph  ->  X  e.  Cat )
140138, 29sseldi 3178 . . . 4  |-  ( ph  ->  Y  e.  Cat )
1411, 139, 140xpccat 13964 . . 3  |-  ( ph  ->  T  e.  Cat )
142 elin 3358 . . 3  |-  ( T  e.  ( U  i^i  Cat )  <->  ( T  e.  U  /\  T  e. 
Cat ) )
143137, 141, 142sylanbrc 645 . 2  |-  ( ph  ->  T  e.  ( U  i^i  Cat ) )
144143, 25eleqtrrd 2360 1  |-  ( ph  ->  T  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   {ctp 3642   <.cop 3643   U.cuni 3827   omcom 4656    X. cxp 4687   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121    ^pm cpm 6773  WUnicwun 8322   1c1 8738   4c4 9797   5c5 9798  ;cdc 10124   ndxcnx 13145   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566  CatCatccatc 13926    X.c cxpc 13942
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-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-catc 13927  df-xpc 13946
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