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Theorem catcxpccl 15334
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 2467 . . . . 5  |-  ( Base `  X )  =  (
Base `  X )
3 eqid 2467 . . . . 5  |-  ( Base `  Y )  =  (
Base `  Y )
4 eqid 2467 . . . . 5  |-  ( Hom  `  X )  =  ( Hom  `  X )
5 eqid 2467 . . . . 5  |-  ( Hom  `  Y )  =  ( Hom  `  Y )
6 eqid 2467 . . . . 5  |-  (comp `  X )  =  (comp `  X )
7 eqid 2467 . . . . 5  |-  (comp `  Y )  =  (comp `  Y )
8 catcxpccl.x . . . . 5  |-  ( ph  ->  X  e.  B )
9 catcxpccl.y . . . . 5  |-  ( ph  ->  Y  e.  B )
10 eqidd 2468 . . . . 5  |-  ( ph  ->  ( ( Base `  X
)  X.  ( Base `  Y ) )  =  ( ( Base `  X
)  X.  ( Base `  Y ) ) )
111, 2, 3xpcbas 15305 . . . . . . 7  |-  ( (
Base `  X )  X.  ( Base `  Y
) )  =  (
Base `  T )
12 eqid 2467 . . . . . . 7  |-  ( Hom  `  T )  =  ( Hom  `  T )
131, 11, 4, 5, 12xpchomfval 15306 . . . . . 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 2468 . . . . 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 15304 . . . 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 14495 . . . . . . 7  |-  Base  = Slot  1
19 catcxpccl.1 . . . . . . . 8  |-  ( ph  ->  om  e.  U )
2017, 19wunndx 14506 . . . . . . 7  |-  ( ph  ->  ndx  e.  U )
2118, 17, 20wunstr 14509 . . . . . 6  |-  ( ph  ->  ( Base `  ndx )  e.  U )
22 inss1 3718 . . . . . . . . 9  |-  ( U  i^i  Cat )  C_  U
23 catcxpccl.c . . . . . . . . . . 11  |-  C  =  (CatCat `  U )
24 catcxpccl.b . . . . . . . . . . 11  |-  B  =  ( Base `  C
)
2523, 24, 17catcbas 15282 . . . . . . . . . 10  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
268, 25eleqtrd 2557 . . . . . . . . 9  |-  ( ph  ->  X  e.  ( U  i^i  Cat ) )
2722, 26sseldi 3502 . . . . . . . 8  |-  ( ph  ->  X  e.  U )
2818, 17, 27wunstr 14509 . . . . . . 7  |-  ( ph  ->  ( Base `  X
)  e.  U )
299, 25eleqtrd 2557 . . . . . . . . 9  |-  ( ph  ->  Y  e.  ( U  i^i  Cat ) )
3022, 29sseldi 3502 . . . . . . . 8  |-  ( ph  ->  Y  e.  U )
3118, 17, 30wunstr 14509 . . . . . . 7  |-  ( ph  ->  ( Base `  Y
)  e.  U )
3217, 28, 31wunxp 9102 . . . . . 6  |-  ( ph  ->  ( ( Base `  X
)  X.  ( Base `  Y ) )  e.  U )
3317, 21, 32wunop 9100 . . . . 5  |-  ( ph  -> 
<. ( Base `  ndx ) ,  ( ( Base `  X )  X.  ( Base `  Y
) ) >.  e.  U
)
34 df-hom 14579 . . . . . . 7  |-  Hom  = Slot ; 1 4
3534, 17, 20wunstr 14509 . . . . . 6  |-  ( ph  ->  ( Hom  `  ndx )  e.  U )
3617, 32, 32wunxp 9102 . . . . . . . 8  |-  ( ph  ->  ( ( ( Base `  X )  X.  ( Base `  Y ) )  X.  ( ( Base `  X )  X.  ( Base `  Y ) ) )  e.  U )
3734, 17, 27wunstr 14509 . . . . . . . . . . . 12  |-  ( ph  ->  ( Hom  `  X
)  e.  U )
3817, 37wunrn 9107 . . . . . . . . . . 11  |-  ( ph  ->  ran  ( Hom  `  X
)  e.  U )
3917, 38wununi 9084 . . . . . . . . . 10  |-  ( ph  ->  U. ran  ( Hom  `  X )  e.  U
)
4034, 17, 30wunstr 14509 . . . . . . . . . . . 12  |-  ( ph  ->  ( Hom  `  Y
)  e.  U )
4117, 40wunrn 9107 . . . . . . . . . . 11  |-  ( ph  ->  ran  ( Hom  `  Y
)  e.  U )
4217, 41wununi 9084 . . . . . . . . . 10  |-  ( ph  ->  U. ran  ( Hom  `  Y )  e.  U
)
4317, 39, 42wunxp 9102 . . . . . . . . 9  |-  ( ph  ->  ( U. ran  ( Hom  `  X )  X. 
U. ran  ( Hom  `  Y ) )  e.  U )
4417, 43wunpw 9085 . . . . . . . 8  |-  ( ph  ->  ~P ( U. ran  ( Hom  `  X )  X.  U. ran  ( Hom  `  Y ) )  e.  U )
45 ovssunirn 6310 . . . . . . . . . . . . 13  |-  ( ( 1st `  u ) ( Hom  `  X
) ( 1st `  v
) )  C_  U. ran  ( Hom  `  X )
46 ovssunirn 6310 . . . . . . . . . . . . 13  |-  ( ( 2nd `  u ) ( Hom  `  Y
) ( 2nd `  v
) )  C_  U. ran  ( Hom  `  Y )
47 xpss12 5108 . . . . . . . . . . . . 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 6309 . . . . . . . . . . . . . 14  |-  ( ( 1st `  u ) ( Hom  `  X
) ( 1st `  v
) )  e.  _V
50 ovex 6309 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  u ) ( Hom  `  Y
) ( 2nd `  v
) )  e.  _V
5149, 50xpex 6588 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  u
) ( Hom  `  X
) ( 1st `  v
) )  X.  (
( 2nd `  u
) ( Hom  `  Y
) ( 2nd `  v
) ) )  e. 
_V
5251elpw 4016 . . . . . . . . . . . 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 2826 . . . . . . . . . 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 2467 . . . . . . . . . . 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 6851 . . . . . . . . . 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 9105 . . . . . . 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 2559 . . . . . 6  |-  ( ph  ->  ( Hom  `  T
)  e.  U )
6117, 35, 60wunop 9100 . . . . 5  |-  ( ph  -> 
<. ( Hom  `  ndx ) ,  ( Hom  `  T ) >.  e.  U
)
62 df-cco 14580 . . . . . . 7  |- comp  = Slot ; 1 5
6362, 17, 20wunstr 14509 . . . . . 6  |-  ( ph  ->  (comp `  ndx )  e.  U )
6417, 36, 32wunxp 9102 . . . . . . 7  |-  ( ph  ->  ( ( ( (
Base `  X )  X.  ( Base `  Y
) )  X.  (
( Base `  X )  X.  ( Base `  Y
) ) )  X.  ( ( Base `  X
)  X.  ( Base `  Y ) ) )  e.  U )
6562, 17, 27wunstr 14509 . . . . . . . . . . . . . 14  |-  ( ph  ->  (comp `  X )  e.  U )
6617, 65wunrn 9107 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  (comp `  X
)  e.  U )
6717, 66wununi 9084 . . . . . . . . . . . 12  |-  ( ph  ->  U. ran  (comp `  X )  e.  U
)
6817, 67wunrn 9107 . . . . . . . . . . 11  |-  ( ph  ->  ran  U. ran  (comp `  X )  e.  U
)
6917, 68wununi 9084 . . . . . . . . . 10  |-  ( ph  ->  U. ran  U. ran  (comp `  X )  e.  U )
7017, 69wunpw 9085 . . . . . . . . 9  |-  ( ph  ->  ~P U. ran  U. ran  (comp `  X )  e.  U )
7162, 17, 30wunstr 14509 . . . . . . . . . . . . . 14  |-  ( ph  ->  (comp `  Y )  e.  U )
7217, 71wunrn 9107 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  (comp `  Y
)  e.  U )
7317, 72wununi 9084 . . . . . . . . . . . 12  |-  ( ph  ->  U. ran  (comp `  Y )  e.  U
)
7417, 73wunrn 9107 . . . . . . . . . . 11  |-  ( ph  ->  ran  U. ran  (comp `  Y )  e.  U
)
7517, 74wununi 9084 . . . . . . . . . 10  |-  ( ph  ->  U. ran  U. ran  (comp `  Y )  e.  U )
7617, 75wunpw 9085 . . . . . . . . 9  |-  ( ph  ->  ~P U. ran  U. ran  (comp `  Y )  e.  U )
7717, 70, 76wunxp 9102 . . . . . . . 8  |-  ( ph  ->  ( ~P U. ran  U.
ran  (comp `  X )  X.  ~P U. ran  U. ran  (comp `  Y )
)  e.  U )
7817, 60wunrn 9107 . . . . . . . . . 10  |-  ( ph  ->  ran  ( Hom  `  T
)  e.  U )
7917, 78wununi 9084 . . . . . . . . 9  |-  ( ph  ->  U. ran  ( Hom  `  T )  e.  U
)
8017, 79, 79wunxp 9102 . . . . . . . 8  |-  ( ph  ->  ( U. ran  ( Hom  `  T )  X. 
U. ran  ( Hom  `  T ) )  e.  U )
8117, 77, 80wunpm 9103 . . . . . . 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 5876 . . . . . . . . . . . . . . . . 17  |-  (comp `  X )  e.  _V
8382rnex 6718 . . . . . . . . . . . . . . . 16  |-  ran  (comp `  X )  e.  _V
8483uniex 6580 . . . . . . . . . . . . . . 15  |-  U. ran  (comp `  X )  e. 
_V
8584rnex 6718 . . . . . . . . . . . . . 14  |-  ran  U. ran  (comp `  X )  e.  _V
8685uniex 6580 . . . . . . . . . . . . 13  |-  U. ran  U.
ran  (comp `  X )  e.  _V
8786pwex 4630 . . . . . . . . . . . 12  |-  ~P U. ran  U. ran  (comp `  X )  e.  _V
88 fvex 5876 . . . . . . . . . . . . . . . . 17  |-  (comp `  Y )  e.  _V
8988rnex 6718 . . . . . . . . . . . . . . . 16  |-  ran  (comp `  Y )  e.  _V
9089uniex 6580 . . . . . . . . . . . . . . 15  |-  U. ran  (comp `  Y )  e. 
_V
9190rnex 6718 . . . . . . . . . . . . . 14  |-  ran  U. ran  (comp `  Y )  e.  _V
9291uniex 6580 . . . . . . . . . . . . 13  |-  U. ran  U.
ran  (comp `  Y )  e.  _V
9392pwex 4630 . . . . . . . . . . . 12  |-  ~P U. ran  U. ran  (comp `  Y )  e.  _V
9487, 93xpex 6588 . . . . . . . . . . 11  |-  ( ~P
U. ran  U. ran  (comp `  X )  X.  ~P U.
ran  U. ran  (comp `  Y ) )  e. 
_V
95 fvex 5876 . . . . . . . . . . . . . 14  |-  ( Hom  `  T )  e.  _V
9695rnex 6718 . . . . . . . . . . . . 13  |-  ran  ( Hom  `  T )  e. 
_V
9796uniex 6580 . . . . . . . . . . . 12  |-  U. ran  ( Hom  `  T )  e.  _V
9897, 97xpex 6588 . . . . . . . . . . 11  |-  ( U. ran  ( Hom  `  T
)  X.  U. ran  ( Hom  `  T )
)  e.  _V
99 ovssunirn 6310 . . . . . . . . . . . . . . . 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 6310 . . . . . . . . . . . . . . . . 17  |-  ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  X )
( 1st `  y
) )  C_  U. ran  (comp `  X )
101 rnss 5231 . . . . . . . . . . . . . . . . 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 4266 . . . . . . . . . . . . . . . . 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 3513 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  g ) ( <. ( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  X )
( 1st `  y
) ) ( 1st `  f ) )  C_  U.
ran  U. ran  (comp `  X )
105 ovex 6309 . . . . . . . . . . . . . . . 16  |-  ( ( 1st `  g ) ( <. ( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  X )
( 1st `  y
) ) ( 1st `  f ) )  e. 
_V
106105elpw 4016 . . . . . . . . . . . . . . 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 6310 . . . . . . . . . . . . . . . 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 6310 . . . . . . . . . . . . . . . . 17  |-  ( <.
( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>. (comp `  Y )
( 2nd `  y
) )  C_  U. ran  (comp `  Y )
110 rnss 5231 . . . . . . . . . . . . . . . . 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 4266 . . . . . . . . . . . . . . . . 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 3513 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  g ) ( <. ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>. (comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) )  C_  U.
ran  U. ran  (comp `  Y )
114 ovex 6309 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  g ) ( <. ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>. (comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) )  e. 
_V
115114elpw 4016 . . . . . . . . . . . . . . 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 5031 . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . 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 2467 . . . . . . . . . . . . 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 6851 . . . . . . . . . . . 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 6310 . . . . . . . . . . . 12  |-  ( ( 2nd `  x ) ( Hom  `  T
) y )  C_  U.
ran  ( Hom  `  T
)
124 fvssunirn 5889 . . . . . . . . . . . 12  |-  ( ( Hom  `  T ) `  x )  C_  U. ran  ( Hom  `  T )
125 xpss12 5108 . . . . . . . . . . . 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 7436 . . . . . . . . . . 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 2826 . . . . . . . . 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 2467 . . . . . . . . . 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 6851 . . . . . . . . 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 9105 . . . . . 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 9100 . . . . 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 9089 . . . 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 2555 . . 3  |-  ( ph  ->  T  e.  U )
138 inss2 3719 . . . . 5  |-  ( U  i^i  Cat )  C_  Cat
139138, 26sseldi 3502 . . . 4  |-  ( ph  ->  X  e.  Cat )
140138, 29sseldi 3502 . . . 4  |-  ( ph  ->  Y  e.  Cat )
1411, 139, 140xpccat 15317 . . 3  |-  ( ph  ->  T  e.  Cat )
142137, 141elind 3688 . 2  |-  ( ph  ->  T  e.  ( U  i^i  Cat ) )
143142, 25eleqtrrd 2558 1  |-  ( ph  ->  T  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   {ctp 4031   <.cop 4033   U.cuni 4245    X. cxp 4997   ran crn 5000   -->wf 5584   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   omcom 6684   1stc1st 6782   2ndc2nd 6783    ^pm cpm 7421  WUnicwun 9078   1c1 9493   4c4 10587   5c5 10588  ;cdc 10976   ndxcnx 14487   Basecbs 14490   Hom chom 14566  compcco 14567   Catccat 14919  CatCatccatc 15279    X.c cxpc 15295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-omul 7135  df-er 7311  df-ec 7313  df-qs 7317  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-wun 9080  df-ni 9250  df-pli 9251  df-mi 9252  df-lti 9253  df-plpq 9286  df-mpq 9287  df-ltpq 9288  df-enq 9289  df-nq 9290  df-erq 9291  df-plq 9292  df-mq 9293  df-1nq 9294  df-rq 9295  df-ltnq 9296  df-np 9359  df-plp 9361  df-ltp 9363  df-enr 9433  df-nr 9434  df-c 9498  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-fz 11673  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-hom 14579  df-cco 14580  df-cat 14923  df-cid 14924  df-catc 15280  df-xpc 15299
This theorem is referenced by: (None)
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