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Theorem catcxpccl 15038
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 2443 . . . . 5  |-  ( Base `  X )  =  (
Base `  X )
3 eqid 2443 . . . . 5  |-  ( Base `  Y )  =  (
Base `  Y )
4 eqid 2443 . . . . 5  |-  ( Hom  `  X )  =  ( Hom  `  X )
5 eqid 2443 . . . . 5  |-  ( Hom  `  Y )  =  ( Hom  `  Y )
6 eqid 2443 . . . . 5  |-  (comp `  X )  =  (comp `  X )
7 eqid 2443 . . . . 5  |-  (comp `  Y )  =  (comp `  Y )
8 catcxpccl.x . . . . 5  |-  ( ph  ->  X  e.  B )
9 catcxpccl.y . . . . 5  |-  ( ph  ->  Y  e.  B )
10 eqidd 2444 . . . . 5  |-  ( ph  ->  ( ( Base `  X
)  X.  ( Base `  Y ) )  =  ( ( Base `  X
)  X.  ( Base `  Y ) ) )
111, 2, 3xpcbas 15009 . . . . . . 7  |-  ( (
Base `  X )  X.  ( Base `  Y
) )  =  (
Base `  T )
12 eqid 2443 . . . . . . 7  |-  ( Hom  `  T )  =  ( Hom  `  T )
131, 11, 4, 5, 12xpchomfval 15010 . . . . . 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 2444 . . . . 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 15008 . . . 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 14200 . . . . . . 7  |-  Base  = Slot  1
19 catcxpccl.1 . . . . . . . 8  |-  ( ph  ->  om  e.  U )
2017, 19wunndx 14211 . . . . . . 7  |-  ( ph  ->  ndx  e.  U )
2118, 17, 20wunstr 14214 . . . . . 6  |-  ( ph  ->  ( Base `  ndx )  e.  U )
22 inss1 3591 . . . . . . . . 9  |-  ( U  i^i  Cat )  C_  U
23 catcxpccl.c . . . . . . . . . . 11  |-  C  =  (CatCat `  U )
24 catcxpccl.b . . . . . . . . . . 11  |-  B  =  ( Base `  C
)
2523, 24, 17catcbas 14986 . . . . . . . . . 10  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
268, 25eleqtrd 2519 . . . . . . . . 9  |-  ( ph  ->  X  e.  ( U  i^i  Cat ) )
2722, 26sseldi 3375 . . . . . . . 8  |-  ( ph  ->  X  e.  U )
2818, 17, 27wunstr 14214 . . . . . . 7  |-  ( ph  ->  ( Base `  X
)  e.  U )
299, 25eleqtrd 2519 . . . . . . . . 9  |-  ( ph  ->  Y  e.  ( U  i^i  Cat ) )
3022, 29sseldi 3375 . . . . . . . 8  |-  ( ph  ->  Y  e.  U )
3118, 17, 30wunstr 14214 . . . . . . 7  |-  ( ph  ->  ( Base `  Y
)  e.  U )
3217, 28, 31wunxp 8912 . . . . . 6  |-  ( ph  ->  ( ( Base `  X
)  X.  ( Base `  Y ) )  e.  U )
3317, 21, 32wunop 8910 . . . . 5  |-  ( ph  -> 
<. ( Base `  ndx ) ,  ( ( Base `  X )  X.  ( Base `  Y
) ) >.  e.  U
)
34 df-hom 14283 . . . . . . 7  |-  Hom  = Slot ; 1 4
3534, 17, 20wunstr 14214 . . . . . 6  |-  ( ph  ->  ( Hom  `  ndx )  e.  U )
3617, 32, 32wunxp 8912 . . . . . . . 8  |-  ( ph  ->  ( ( ( Base `  X )  X.  ( Base `  Y ) )  X.  ( ( Base `  X )  X.  ( Base `  Y ) ) )  e.  U )
3734, 17, 27wunstr 14214 . . . . . . . . . . . 12  |-  ( ph  ->  ( Hom  `  X
)  e.  U )
3817, 37wunrn 8917 . . . . . . . . . . 11  |-  ( ph  ->  ran  ( Hom  `  X
)  e.  U )
3917, 38wununi 8894 . . . . . . . . . 10  |-  ( ph  ->  U. ran  ( Hom  `  X )  e.  U
)
4034, 17, 30wunstr 14214 . . . . . . . . . . . 12  |-  ( ph  ->  ( Hom  `  Y
)  e.  U )
4117, 40wunrn 8917 . . . . . . . . . . 11  |-  ( ph  ->  ran  ( Hom  `  Y
)  e.  U )
4217, 41wununi 8894 . . . . . . . . . 10  |-  ( ph  ->  U. ran  ( Hom  `  Y )  e.  U
)
4317, 39, 42wunxp 8912 . . . . . . . . 9  |-  ( ph  ->  ( U. ran  ( Hom  `  X )  X. 
U. ran  ( Hom  `  Y ) )  e.  U )
4417, 43wunpw 8895 . . . . . . . 8  |-  ( ph  ->  ~P ( U. ran  ( Hom  `  X )  X.  U. ran  ( Hom  `  Y ) )  e.  U )
45 ovssunirn 6138 . . . . . . . . . . . . 13  |-  ( ( 1st `  u ) ( Hom  `  X
) ( 1st `  v
) )  C_  U. ran  ( Hom  `  X )
46 ovssunirn 6138 . . . . . . . . . . . . 13  |-  ( ( 2nd `  u ) ( Hom  `  Y
) ( 2nd `  v
) )  C_  U. ran  ( Hom  `  Y )
47 xpss12 4966 . . . . . . . . . . . . 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 672 . . . . . . . . . . . 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 6137 . . . . . . . . . . . . . 14  |-  ( ( 1st `  u ) ( Hom  `  X
) ( 1st `  v
) )  e.  _V
50 ovex 6137 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  u ) ( Hom  `  Y
) ( 2nd `  v
) )  e.  _V
5149, 50xpex 6529 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  u
) ( Hom  `  X
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  Y
) ( 2nd `  v
) ) )  e. 
_V
5251elpw 3887 . . . . . . . . . . . 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 209 . . . . . . . . . . 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 2805 . . . . . . . . . 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 2443 . . . . . . . . . . 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 6662 . . . . . . . . . 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 208 . . . . . . . . 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 8915 . . . . . . 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 2527 . . . . . 6  |-  ( ph  ->  ( Hom  `  T
)  e.  U )
6117, 35, 60wunop 8910 . . . . 5  |-  ( ph  -> 
<. ( Hom  `  ndx ) ,  ( Hom  `  T ) >.  e.  U
)
62 df-cco 14284 . . . . . . 7  |- comp  = Slot ; 1 5
6362, 17, 20wunstr 14214 . . . . . 6  |-  ( ph  ->  (comp `  ndx )  e.  U )
6417, 36, 32wunxp 8912 . . . . . . 7  |-  ( ph  ->  ( ( ( (
Base `  X )  X.  ( Base `  Y
) )  X.  (
( Base `  X )  X.  ( Base `  Y
) ) )  X.  ( ( Base `  X
)  X.  ( Base `  Y ) ) )  e.  U )
6562, 17, 27wunstr 14214 . . . . . . . . . . . . . 14  |-  ( ph  ->  (comp `  X )  e.  U )
6617, 65wunrn 8917 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  (comp `  X
)  e.  U )
6717, 66wununi 8894 . . . . . . . . . . . 12  |-  ( ph  ->  U. ran  (comp `  X )  e.  U
)
6817, 67wunrn 8917 . . . . . . . . . . 11  |-  ( ph  ->  ran  U. ran  (comp `  X )  e.  U
)
6917, 68wununi 8894 . . . . . . . . . 10  |-  ( ph  ->  U. ran  U. ran  (comp `  X )  e.  U )
7017, 69wunpw 8895 . . . . . . . . 9  |-  ( ph  ->  ~P U. ran  U. ran  (comp `  X )  e.  U )
7162, 17, 30wunstr 14214 . . . . . . . . . . . . . 14  |-  ( ph  ->  (comp `  Y )  e.  U )
7217, 71wunrn 8917 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  (comp `  Y
)  e.  U )
7317, 72wununi 8894 . . . . . . . . . . . 12  |-  ( ph  ->  U. ran  (comp `  Y )  e.  U
)
7417, 73wunrn 8917 . . . . . . . . . . 11  |-  ( ph  ->  ran  U. ran  (comp `  Y )  e.  U
)
7517, 74wununi 8894 . . . . . . . . . 10  |-  ( ph  ->  U. ran  U. ran  (comp `  Y )  e.  U )
7617, 75wunpw 8895 . . . . . . . . 9  |-  ( ph  ->  ~P U. ran  U. ran  (comp `  Y )  e.  U )
7717, 70, 76wunxp 8912 . . . . . . . 8  |-  ( ph  ->  ( ~P U. ran  U.
ran  (comp `  X )  X.  ~P U. ran  U. ran  (comp `  Y )
)  e.  U )
7817, 60wunrn 8917 . . . . . . . . . 10  |-  ( ph  ->  ran  ( Hom  `  T
)  e.  U )
7917, 78wununi 8894 . . . . . . . . 9  |-  ( ph  ->  U. ran  ( Hom  `  T )  e.  U
)
8017, 79, 79wunxp 8912 . . . . . . . 8  |-  ( ph  ->  ( U. ran  ( Hom  `  T )  X. 
U. ran  ( Hom  `  T ) )  e.  U )
8117, 77, 80wunpm 8913 . . . . . . 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 5722 . . . . . . . . . . . . . . . . 17  |-  (comp `  X )  e.  _V
8382rnex 6533 . . . . . . . . . . . . . . . 16  |-  ran  (comp `  X )  e.  _V
8483uniex 6397 . . . . . . . . . . . . . . 15  |-  U. ran  (comp `  X )  e. 
_V
8584rnex 6533 . . . . . . . . . . . . . 14  |-  ran  U. ran  (comp `  X )  e.  _V
8685uniex 6397 . . . . . . . . . . . . 13  |-  U. ran  U.
ran  (comp `  X )  e.  _V
8786pwex 4496 . . . . . . . . . . . 12  |-  ~P U. ran  U. ran  (comp `  X )  e.  _V
88 fvex 5722 . . . . . . . . . . . . . . . . 17  |-  (comp `  Y )  e.  _V
8988rnex 6533 . . . . . . . . . . . . . . . 16  |-  ran  (comp `  Y )  e.  _V
9089uniex 6397 . . . . . . . . . . . . . . 15  |-  U. ran  (comp `  Y )  e. 
_V
9190rnex 6533 . . . . . . . . . . . . . 14  |-  ran  U. ran  (comp `  Y )  e.  _V
9291uniex 6397 . . . . . . . . . . . . 13  |-  U. ran  U.
ran  (comp `  Y )  e.  _V
9392pwex 4496 . . . . . . . . . . . 12  |-  ~P U. ran  U. ran  (comp `  Y )  e.  _V
9487, 93xpex 6529 . . . . . . . . . . 11  |-  ( ~P
U. ran  U. ran  (comp `  X )  X.  ~P U.
ran  U. ran  (comp `  Y ) )  e. 
_V
95 fvex 5722 . . . . . . . . . . . . . 14  |-  ( Hom  `  T )  e.  _V
9695rnex 6533 . . . . . . . . . . . . 13  |-  ran  ( Hom  `  T )  e. 
_V
9796uniex 6397 . . . . . . . . . . . 12  |-  U. ran  ( Hom  `  T )  e.  _V
9897, 97xpex 6529 . . . . . . . . . . 11  |-  ( U. ran  ( Hom  `  T
)  X.  U. ran  ( Hom  `  T )
)  e.  _V
99 ovssunirn 6138 . . . . . . . . . . . . . . . 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 6138 . . . . . . . . . . . . . . . . 17  |-  ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  X )
( 1st `  y
) )  C_  U. ran  (comp `  X )
101 rnss 5089 . . . . . . . . . . . . . . . . 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 4133 . . . . . . . . . . . . . . . . 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 3386 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  g ) ( <. ( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  X )
( 1st `  y
) ) ( 1st `  f ) )  C_  U.
ran  U. ran  (comp `  X )
105 ovex 6137 . . . . . . . . . . . . . . . 16  |-  ( ( 1st `  g ) ( <. ( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  X )
( 1st `  y
) ) ( 1st `  f ) )  e. 
_V
106105elpw 3887 . . . . . . . . . . . . . . 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 209 . . . . . . . . . . . . . 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 6138 . . . . . . . . . . . . . . . 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 6138 . . . . . . . . . . . . . . . . 17  |-  ( <.
( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>. (comp `  Y )
( 2nd `  y
) )  C_  U. ran  (comp `  Y )
110 rnss 5089 . . . . . . . . . . . . . . . . 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 4133 . . . . . . . . . . . . . . . . 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 3386 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  g ) ( <. ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>. (comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) )  C_  U.
ran  U. ran  (comp `  Y )
114 ovex 6137 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  g ) ( <. ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>. (comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) )  e. 
_V
115114elpw 3887 . . . . . . . . . . . . . . 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 209 . . . . . . . . . . . . . 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 4892 . . . . . . . . . . . . . 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 672 . . . . . . . . . . . . 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 2805 . . . . . . . . . . . 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 2443 . . . . . . . . . . . . 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 6662 . . . . . . . . . . . 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 208 . . . . . . . . . . 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 6138 . . . . . . . . . . . 12  |-  ( ( 2nd `  x ) ( Hom  `  T
) y )  C_  U.
ran  ( Hom  `  T
)
124 fvssunirn 5734 . . . . . . . . . . . 12  |-  ( ( Hom  `  T ) `  x )  C_  U. ran  ( Hom  `  T )
125 xpss12 4966 . . . . . . . . . . . 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 672 . . . . . . . . . . 11  |-  ( ( ( 2nd `  x
) ( Hom  `  T
) y )  X.  ( ( Hom  `  T
) `  x )
)  C_  ( U. ran  ( Hom  `  T
)  X.  U. ran  ( Hom  `  T )
)
127 elpm2r 7251 . . . . . . . . . . 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 673 . . . . . . . . . 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 2805 . . . . . . . . 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 2443 . . . . . . . . . 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 6662 . . . . . . . . 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 208 . . . . . . . 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 8915 . . . . . 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 8910 . . . . 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 8899 . . . 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 2517 . . 3  |-  ( ph  ->  T  e.  U )
138 inss2 3592 . . . . 5  |-  ( U  i^i  Cat )  C_  Cat
139138, 26sseldi 3375 . . . 4  |-  ( ph  ->  X  e.  Cat )
140138, 29sseldi 3375 . . . 4  |-  ( ph  ->  Y  e.  Cat )
1411, 139, 140xpccat 15021 . . 3  |-  ( ph  ->  T  e.  Cat )
142137, 141elind 3561 . 2  |-  ( ph  ->  T  e.  ( U  i^i  Cat ) )
143142, 25eleqtrrd 2520 1  |-  ( ph  ->  T  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   A.wral 2736   _Vcvv 2993    i^i cin 3348    C_ wss 3349   ~Pcpw 3881   {ctp 3902   <.cop 3904   U.cuni 4112    X. cxp 4859   ran crn 4862   -->wf 5435   ` cfv 5439  (class class class)co 6112    e. cmpt2 6114   omcom 6497   1stc1st 6596   2ndc2nd 6597    ^pm cpm 7236  WUnicwun 8888   1c1 9304   4c4 10394   5c5 10395  ;cdc 10776   ndxcnx 14192   Basecbs 14195   Hom chom 14270  compcco 14271   Catccat 14623  CatCatccatc 14983    X.c cxpc 14999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-inf2 7868  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-omul 6946  df-er 7122  df-ec 7124  df-qs 7128  df-map 7237  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-wun 8890  df-ni 9062  df-pli 9063  df-mi 9064  df-lti 9065  df-plpq 9098  df-mpq 9099  df-ltpq 9100  df-enq 9101  df-nq 9102  df-erq 9103  df-plq 9104  df-mq 9105  df-1nq 9106  df-rq 9107  df-ltnq 9108  df-np 9171  df-plp 9173  df-ltp 9175  df-enr 9247  df-nr 9248  df-c 9309  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-fz 11459  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-hom 14283  df-cco 14284  df-cat 14627  df-cid 14628  df-catc 14984  df-xpc 15003
This theorem is referenced by: (None)
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