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Theorem catcxpccl 14309
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 2438 . . . . 5  |-  ( Base `  X )  =  (
Base `  X )
3 eqid 2438 . . . . 5  |-  ( Base `  Y )  =  (
Base `  Y )
4 eqid 2438 . . . . 5  |-  (  Hom  `  X )  =  (  Hom  `  X )
5 eqid 2438 . . . . 5  |-  (  Hom  `  Y )  =  (  Hom  `  Y )
6 eqid 2438 . . . . 5  |-  (comp `  X )  =  (comp `  X )
7 eqid 2438 . . . . 5  |-  (comp `  Y )  =  (comp `  Y )
8 catcxpccl.x . . . . 5  |-  ( ph  ->  X  e.  B )
9 catcxpccl.y . . . . 5  |-  ( ph  ->  Y  e.  B )
10 eqidd 2439 . . . . 5  |-  ( ph  ->  ( ( Base `  X
)  X.  ( Base `  Y ) )  =  ( ( Base `  X
)  X.  ( Base `  Y ) ) )
111, 2, 3xpcbas 14280 . . . . . . 7  |-  ( (
Base `  X )  X.  ( Base `  Y
) )  =  (
Base `  T )
12 eqid 2438 . . . . . . 7  |-  (  Hom  `  T )  =  (  Hom  `  T )
131, 11, 4, 5, 12xpchomfval 14281 . . . . . 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 11 . . . . 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 2439 . . . . 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 14279 . . . 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 13479 . . . . . . 7  |-  Base  = Slot  1
19 catcxpccl.1 . . . . . . . 8  |-  ( ph  ->  om  e.  U )
2017, 19wunndx 13490 . . . . . . 7  |-  ( ph  ->  ndx  e.  U )
2118, 17, 20wunstr 13493 . . . . . 6  |-  ( ph  ->  ( Base `  ndx )  e.  U )
22 inss1 3563 . . . . . . . . 9  |-  ( U  i^i  Cat )  C_  U
23 catcxpccl.c . . . . . . . . . . 11  |-  C  =  (CatCat `  U )
24 catcxpccl.b . . . . . . . . . . 11  |-  B  =  ( Base `  C
)
2523, 24, 17catcbas 14257 . . . . . . . . . 10  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
268, 25eleqtrd 2514 . . . . . . . . 9  |-  ( ph  ->  X  e.  ( U  i^i  Cat ) )
2722, 26sseldi 3348 . . . . . . . 8  |-  ( ph  ->  X  e.  U )
2818, 17, 27wunstr 13493 . . . . . . 7  |-  ( ph  ->  ( Base `  X
)  e.  U )
299, 25eleqtrd 2514 . . . . . . . . 9  |-  ( ph  ->  Y  e.  ( U  i^i  Cat ) )
3022, 29sseldi 3348 . . . . . . . 8  |-  ( ph  ->  Y  e.  U )
3118, 17, 30wunstr 13493 . . . . . . 7  |-  ( ph  ->  ( Base `  Y
)  e.  U )
3217, 28, 31wunxp 8604 . . . . . 6  |-  ( ph  ->  ( ( Base `  X
)  X.  ( Base `  Y ) )  e.  U )
3317, 21, 32wunop 8602 . . . . 5  |-  ( ph  -> 
<. ( Base `  ndx ) ,  ( ( Base `  X )  X.  ( Base `  Y
) ) >.  e.  U
)
34 df-hom 13558 . . . . . . 7  |-  Hom  = Slot ; 1 4
3534, 17, 20wunstr 13493 . . . . . 6  |-  ( ph  ->  (  Hom  `  ndx )  e.  U )
3617, 32, 32wunxp 8604 . . . . . . . 8  |-  ( ph  ->  ( ( ( Base `  X )  X.  ( Base `  Y ) )  X.  ( ( Base `  X )  X.  ( Base `  Y ) ) )  e.  U )
3734, 17, 27wunstr 13493 . . . . . . . . . . . 12  |-  ( ph  ->  (  Hom  `  X
)  e.  U )
3817, 37wunrn 8609 . . . . . . . . . . 11  |-  ( ph  ->  ran  (  Hom  `  X
)  e.  U )
3917, 38wununi 8586 . . . . . . . . . 10  |-  ( ph  ->  U. ran  (  Hom  `  X )  e.  U
)
4034, 17, 30wunstr 13493 . . . . . . . . . . . 12  |-  ( ph  ->  (  Hom  `  Y
)  e.  U )
4117, 40wunrn 8609 . . . . . . . . . . 11  |-  ( ph  ->  ran  (  Hom  `  Y
)  e.  U )
4217, 41wununi 8586 . . . . . . . . . 10  |-  ( ph  ->  U. ran  (  Hom  `  Y )  e.  U
)
4317, 39, 42wunxp 8604 . . . . . . . . 9  |-  ( ph  ->  ( U. ran  (  Hom  `  X )  X. 
U. ran  (  Hom  `  Y ) )  e.  U )
4417, 43wunpw 8587 . . . . . . . 8  |-  ( ph  ->  ~P ( U. ran  (  Hom  `  X )  X.  U. ran  (  Hom  `  Y ) )  e.  U )
45 ovssunirn 6110 . . . . . . . . . . . . 13  |-  ( ( 1st `  u ) (  Hom  `  X
) ( 1st `  v
) )  C_  U. ran  (  Hom  `  X )
46 ovssunirn 6110 . . . . . . . . . . . . 13  |-  ( ( 2nd `  u ) (  Hom  `  Y
) ( 2nd `  v
) )  C_  U. ran  (  Hom  `  Y )
47 xpss12 4984 . . . . . . . . . . . . 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 655 . . . . . . . . . . . 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 6109 . . . . . . . . . . . . . 14  |-  ( ( 1st `  u ) (  Hom  `  X
) ( 1st `  v
) )  e.  _V
50 ovex 6109 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  u ) (  Hom  `  Y
) ( 2nd `  v
) )  e.  _V
5149, 50xpex 4993 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  u
) (  Hom  `  X
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  Y
) ( 2nd `  v
) ) )  e. 
_V
5251elpw 3807 . . . . . . . . . . . 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 202 . . . . . . . . . . 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 2776 . . . . . . . . . 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 2438 . . . . . . . . . . 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 6421 . . . . . . . . . 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 201 . . . . . . . . 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 11 . . . . . . . 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 8607 . . . . . . 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 2522 . . . . . 6  |-  ( ph  ->  (  Hom  `  T
)  e.  U )
6117, 35, 60wunop 8602 . . . . 5  |-  ( ph  -> 
<. (  Hom  `  ndx ) ,  (  Hom  `  T ) >.  e.  U
)
62 df-cco 13559 . . . . . . 7  |- comp  = Slot ; 1 5
6362, 17, 20wunstr 13493 . . . . . 6  |-  ( ph  ->  (comp `  ndx )  e.  U )
6417, 36, 32wunxp 8604 . . . . . . 7  |-  ( ph  ->  ( ( ( (
Base `  X )  X.  ( Base `  Y
) )  X.  (
( Base `  X )  X.  ( Base `  Y
) ) )  X.  ( ( Base `  X
)  X.  ( Base `  Y ) ) )  e.  U )
6562, 17, 27wunstr 13493 . . . . . . . . . . . . . 14  |-  ( ph  ->  (comp `  X )  e.  U )
6617, 65wunrn 8609 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  (comp `  X
)  e.  U )
6717, 66wununi 8586 . . . . . . . . . . . 12  |-  ( ph  ->  U. ran  (comp `  X )  e.  U
)
6817, 67wunrn 8609 . . . . . . . . . . 11  |-  ( ph  ->  ran  U. ran  (comp `  X )  e.  U
)
6917, 68wununi 8586 . . . . . . . . . 10  |-  ( ph  ->  U. ran  U. ran  (comp `  X )  e.  U )
7017, 69wunpw 8587 . . . . . . . . 9  |-  ( ph  ->  ~P U. ran  U. ran  (comp `  X )  e.  U )
7162, 17, 30wunstr 13493 . . . . . . . . . . . . . 14  |-  ( ph  ->  (comp `  Y )  e.  U )
7217, 71wunrn 8609 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  (comp `  Y
)  e.  U )
7317, 72wununi 8586 . . . . . . . . . . . 12  |-  ( ph  ->  U. ran  (comp `  Y )  e.  U
)
7417, 73wunrn 8609 . . . . . . . . . . 11  |-  ( ph  ->  ran  U. ran  (comp `  Y )  e.  U
)
7517, 74wununi 8586 . . . . . . . . . 10  |-  ( ph  ->  U. ran  U. ran  (comp `  Y )  e.  U )
7617, 75wunpw 8587 . . . . . . . . 9  |-  ( ph  ->  ~P U. ran  U. ran  (comp `  Y )  e.  U )
7717, 70, 76wunxp 8604 . . . . . . . 8  |-  ( ph  ->  ( ~P U. ran  U.
ran  (comp `  X )  X.  ~P U. ran  U. ran  (comp `  Y )
)  e.  U )
7817, 60wunrn 8609 . . . . . . . . . 10  |-  ( ph  ->  ran  (  Hom  `  T
)  e.  U )
7917, 78wununi 8586 . . . . . . . . 9  |-  ( ph  ->  U. ran  (  Hom  `  T )  e.  U
)
8017, 79, 79wunxp 8604 . . . . . . . 8  |-  ( ph  ->  ( U. ran  (  Hom  `  T )  X. 
U. ran  (  Hom  `  T ) )  e.  U )
8117, 77, 80wunpm 8605 . . . . . . 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 5745 . . . . . . . . . . . . . . . . 17  |-  (comp `  X )  e.  _V
8382rnex 5136 . . . . . . . . . . . . . . . 16  |-  ran  (comp `  X )  e.  _V
8483uniex 4708 . . . . . . . . . . . . . . 15  |-  U. ran  (comp `  X )  e. 
_V
8584rnex 5136 . . . . . . . . . . . . . 14  |-  ran  U. ran  (comp `  X )  e.  _V
8685uniex 4708 . . . . . . . . . . . . 13  |-  U. ran  U.
ran  (comp `  X )  e.  _V
8786pwex 4385 . . . . . . . . . . . 12  |-  ~P U. ran  U. ran  (comp `  X )  e.  _V
88 fvex 5745 . . . . . . . . . . . . . . . . 17  |-  (comp `  Y )  e.  _V
8988rnex 5136 . . . . . . . . . . . . . . . 16  |-  ran  (comp `  Y )  e.  _V
9089uniex 4708 . . . . . . . . . . . . . . 15  |-  U. ran  (comp `  Y )  e. 
_V
9190rnex 5136 . . . . . . . . . . . . . 14  |-  ran  U. ran  (comp `  Y )  e.  _V
9291uniex 4708 . . . . . . . . . . . . 13  |-  U. ran  U.
ran  (comp `  Y )  e.  _V
9392pwex 4385 . . . . . . . . . . . 12  |-  ~P U. ran  U. ran  (comp `  Y )  e.  _V
9487, 93xpex 4993 . . . . . . . . . . 11  |-  ( ~P
U. ran  U. ran  (comp `  X )  X.  ~P U.
ran  U. ran  (comp `  Y ) )  e. 
_V
95 fvex 5745 . . . . . . . . . . . . . 14  |-  (  Hom  `  T )  e.  _V
9695rnex 5136 . . . . . . . . . . . . 13  |-  ran  (  Hom  `  T )  e. 
_V
9796uniex 4708 . . . . . . . . . . . 12  |-  U. ran  (  Hom  `  T )  e.  _V
9897, 97xpex 4993 . . . . . . . . . . 11  |-  ( U. ran  (  Hom  `  T
)  X.  U. ran  (  Hom  `  T )
)  e.  _V
99 ovssunirn 6110 . . . . . . . . . . . . . . . 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 6110 . . . . . . . . . . . . . . . . 17  |-  ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  X )
( 1st `  y
) )  C_  U. ran  (comp `  X )
101 rnss 5101 . . . . . . . . . . . . . . . . 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 4038 . . . . . . . . . . . . . . . . 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 10 . . . . . . . . . . . . . . . 16  |-  U. ran  ( <. ( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  X )
( 1st `  y
) )  C_  U. ran  U.
ran  (comp `  X )
10499, 103sstri 3359 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  g ) ( <. ( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  X )
( 1st `  y
) ) ( 1st `  f ) )  C_  U.
ran  U. ran  (comp `  X )
105 ovex 6109 . . . . . . . . . . . . . . . 16  |-  ( ( 1st `  g ) ( <. ( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  X )
( 1st `  y
) ) ( 1st `  f ) )  e. 
_V
106105elpw 3807 . . . . . . . . . . . . . . 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 202 . . . . . . . . . . . . . 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 6110 . . . . . . . . . . . . . . . 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 6110 . . . . . . . . . . . . . . . . 17  |-  ( <.
( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>. (comp `  Y )
( 2nd `  y
) )  C_  U. ran  (comp `  Y )
110 rnss 5101 . . . . . . . . . . . . . . . . 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 4038 . . . . . . . . . . . . . . . . 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 10 . . . . . . . . . . . . . . . 16  |-  U. ran  ( <. ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>. (comp `  Y )
( 2nd `  y
) )  C_  U. ran  U.
ran  (comp `  Y )
113108, 112sstri 3359 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  g ) ( <. ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>. (comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) )  C_  U.
ran  U. ran  (comp `  Y )
114 ovex 6109 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  g ) ( <. ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>. (comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) )  e. 
_V
115114elpw 3807 . . . . . . . . . . . . . . 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 202 . . . . . . . . . . . . . 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 4913 . . . . . . . . . . . . . 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 655 . . . . . . . . . . . . 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 2776 . . . . . . . . . . . 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 2438 . . . . . . . . . . . . 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 6421 . . . . . . . . . . . 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 201 . . . . . . . . . . 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 6110 . . . . . . . . . . . 12  |-  ( ( 2nd `  x ) (  Hom  `  T
) y )  C_  U.
ran  (  Hom  `  T
)
124 fvssunirn 5757 . . . . . . . . . . . 12  |-  ( (  Hom  `  T ) `  x )  C_  U. ran  (  Hom  `  T )
125 xpss12 4984 . . . . . . . . . . . 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 655 . . . . . . . . . . 11  |-  ( ( ( 2nd `  x
) (  Hom  `  T
) y )  X.  ( (  Hom  `  T
) `  x )
)  C_  ( U. ran  (  Hom  `  T
)  X.  U. ran  (  Hom  `  T )
)
127 elpm2r 7037 . . . . . . . . . . 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 656 . . . . . . . . . 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 2776 . . . . . . . . 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 2438 . . . . . . . . . 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 6421 . . . . . . . . 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 201 . . . . . . . 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 11 . . . . . . 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 8607 . . . . . 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 8602 . . . . 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 8591 . . . 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 2512 . . 3  |-  ( ph  ->  T  e.  U )
138 inss2 3564 . . . . 5  |-  ( U  i^i  Cat )  C_  Cat
139138, 26sseldi 3348 . . . 4  |-  ( ph  ->  X  e.  Cat )
140138, 29sseldi 3348 . . . 4  |-  ( ph  ->  Y  e.  Cat )
1411, 139, 140xpccat 14292 . . 3  |-  ( ph  ->  T  e.  Cat )
142 elin 3532 . . 3  |-  ( T  e.  ( U  i^i  Cat )  <->  ( T  e.  U  /\  T  e. 
Cat ) )
143137, 141, 142sylanbrc 647 . 2  |-  ( ph  ->  T  e.  ( U  i^i  Cat ) )
144143, 25eleqtrrd 2515 1  |-  ( ph  ->  T  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    i^i cin 3321    C_ wss 3322   ~Pcpw 3801   {ctp 3818   <.cop 3819   U.cuni 4017   omcom 4848    X. cxp 4879   ran crn 4882   -->wf 5453   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   1stc1st 6350   2ndc2nd 6351    ^pm cpm 7022  WUnicwun 8580   1c1 8996   4c4 10056   5c5 10057  ;cdc 10387   ndxcnx 13471   Basecbs 13474    Hom chom 13545  compcco 13546   Catccat 13894  CatCatccatc 14254    X.c cxpc 14270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-omul 6732  df-er 6908  df-ec 6910  df-qs 6914  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-wun 8582  df-ni 8754  df-pli 8755  df-mi 8756  df-lti 8757  df-plpq 8790  df-mpq 8791  df-ltpq 8792  df-enq 8793  df-nq 8794  df-erq 8795  df-plq 8796  df-mq 8797  df-1nq 8798  df-rq 8799  df-ltnq 8800  df-np 8863  df-plp 8865  df-ltp 8867  df-enr 8939  df-nr 8940  df-c 9001  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-fz 11049  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-hom 13558  df-cco 13559  df-cat 13898  df-cid 13899  df-catc 14255  df-xpc 14274
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