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Theorem catcxpccl 14263
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 2408 . . . . 5  |-  ( Base `  X )  =  (
Base `  X )
3 eqid 2408 . . . . 5  |-  ( Base `  Y )  =  (
Base `  Y )
4 eqid 2408 . . . . 5  |-  (  Hom  `  X )  =  (  Hom  `  X )
5 eqid 2408 . . . . 5  |-  (  Hom  `  Y )  =  (  Hom  `  Y )
6 eqid 2408 . . . . 5  |-  (comp `  X )  =  (comp `  X )
7 eqid 2408 . . . . 5  |-  (comp `  Y )  =  (comp `  Y )
8 catcxpccl.x . . . . 5  |-  ( ph  ->  X  e.  B )
9 catcxpccl.y . . . . 5  |-  ( ph  ->  Y  e.  B )
10 eqidd 2409 . . . . 5  |-  ( ph  ->  ( ( Base `  X
)  X.  ( Base `  Y ) )  =  ( ( Base `  X
)  X.  ( Base `  Y ) ) )
111, 2, 3xpcbas 14234 . . . . . . 7  |-  ( (
Base `  X )  X.  ( Base `  Y
) )  =  (
Base `  T )
12 eqid 2408 . . . . . . 7  |-  (  Hom  `  T )  =  (  Hom  `  T )
131, 11, 4, 5, 12xpchomfval 14235 . . . . . 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 2409 . . . . 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 14233 . . . 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 13433 . . . . . . 7  |-  Base  = Slot  1
19 catcxpccl.1 . . . . . . . 8  |-  ( ph  ->  om  e.  U )
2017, 19wunndx 13444 . . . . . . 7  |-  ( ph  ->  ndx  e.  U )
2118, 17, 20wunstr 13447 . . . . . 6  |-  ( ph  ->  ( Base `  ndx )  e.  U )
22 inss1 3525 . . . . . . . . 9  |-  ( U  i^i  Cat )  C_  U
23 catcxpccl.c . . . . . . . . . . 11  |-  C  =  (CatCat `  U )
24 catcxpccl.b . . . . . . . . . . 11  |-  B  =  ( Base `  C
)
2523, 24, 17catcbas 14211 . . . . . . . . . 10  |-  ( ph  ->  B  =  ( U  i^i  Cat ) )
268, 25eleqtrd 2484 . . . . . . . . 9  |-  ( ph  ->  X  e.  ( U  i^i  Cat ) )
2722, 26sseldi 3310 . . . . . . . 8  |-  ( ph  ->  X  e.  U )
2818, 17, 27wunstr 13447 . . . . . . 7  |-  ( ph  ->  ( Base `  X
)  e.  U )
299, 25eleqtrd 2484 . . . . . . . . 9  |-  ( ph  ->  Y  e.  ( U  i^i  Cat ) )
3022, 29sseldi 3310 . . . . . . . 8  |-  ( ph  ->  Y  e.  U )
3118, 17, 30wunstr 13447 . . . . . . 7  |-  ( ph  ->  ( Base `  Y
)  e.  U )
3217, 28, 31wunxp 8559 . . . . . 6  |-  ( ph  ->  ( ( Base `  X
)  X.  ( Base `  Y ) )  e.  U )
3317, 21, 32wunop 8557 . . . . 5  |-  ( ph  -> 
<. ( Base `  ndx ) ,  ( ( Base `  X )  X.  ( Base `  Y
) ) >.  e.  U
)
34 df-hom 13512 . . . . . . 7  |-  Hom  = Slot ; 1 4
3534, 17, 20wunstr 13447 . . . . . 6  |-  ( ph  ->  (  Hom  `  ndx )  e.  U )
3617, 32, 32wunxp 8559 . . . . . . . 8  |-  ( ph  ->  ( ( ( Base `  X )  X.  ( Base `  Y ) )  X.  ( ( Base `  X )  X.  ( Base `  Y ) ) )  e.  U )
3734, 17, 27wunstr 13447 . . . . . . . . . . . 12  |-  ( ph  ->  (  Hom  `  X
)  e.  U )
3817, 37wunrn 8564 . . . . . . . . . . 11  |-  ( ph  ->  ran  (  Hom  `  X
)  e.  U )
3917, 38wununi 8541 . . . . . . . . . 10  |-  ( ph  ->  U. ran  (  Hom  `  X )  e.  U
)
4034, 17, 30wunstr 13447 . . . . . . . . . . . 12  |-  ( ph  ->  (  Hom  `  Y
)  e.  U )
4117, 40wunrn 8564 . . . . . . . . . . 11  |-  ( ph  ->  ran  (  Hom  `  Y
)  e.  U )
4217, 41wununi 8541 . . . . . . . . . 10  |-  ( ph  ->  U. ran  (  Hom  `  Y )  e.  U
)
4317, 39, 42wunxp 8559 . . . . . . . . 9  |-  ( ph  ->  ( U. ran  (  Hom  `  X )  X. 
U. ran  (  Hom  `  Y ) )  e.  U )
4417, 43wunpw 8542 . . . . . . . 8  |-  ( ph  ->  ~P ( U. ran  (  Hom  `  X )  X.  U. ran  (  Hom  `  Y ) )  e.  U )
45 ovssunirn 6070 . . . . . . . . . . . . 13  |-  ( ( 1st `  u ) (  Hom  `  X
) ( 1st `  v
) )  C_  U. ran  (  Hom  `  X )
46 ovssunirn 6070 . . . . . . . . . . . . 13  |-  ( ( 2nd `  u ) (  Hom  `  Y
) ( 2nd `  v
) )  C_  U. ran  (  Hom  `  Y )
47 xpss12 4944 . . . . . . . . . . . . 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 654 . . . . . . . . . . . 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 6069 . . . . . . . . . . . . . 14  |-  ( ( 1st `  u ) (  Hom  `  X
) ( 1st `  v
) )  e.  _V
50 ovex 6069 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  u ) (  Hom  `  Y
) ( 2nd `  v
) )  e.  _V
5149, 50xpex 4953 . . . . . . . . . . . . 13  |-  ( ( ( 1st `  u
) (  Hom  `  X
) ( 1st `  v
) )  X.  (
( 2nd `  u
) (  Hom  `  Y
) ( 2nd `  v
) ) )  e. 
_V
5251elpw 3769 . . . . . . . . . . . 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 201 . . . . . . . . . . 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 2738 . . . . . . . . . 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 2408 . . . . . . . . . . 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 6381 . . . . . . . . . 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 200 . . . . . . . . 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 8562 . . . . . . 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 2492 . . . . . 6  |-  ( ph  ->  (  Hom  `  T
)  e.  U )
6117, 35, 60wunop 8557 . . . . 5  |-  ( ph  -> 
<. (  Hom  `  ndx ) ,  (  Hom  `  T ) >.  e.  U
)
62 df-cco 13513 . . . . . . 7  |- comp  = Slot ; 1 5
6362, 17, 20wunstr 13447 . . . . . 6  |-  ( ph  ->  (comp `  ndx )  e.  U )
6417, 36, 32wunxp 8559 . . . . . . 7  |-  ( ph  ->  ( ( ( (
Base `  X )  X.  ( Base `  Y
) )  X.  (
( Base `  X )  X.  ( Base `  Y
) ) )  X.  ( ( Base `  X
)  X.  ( Base `  Y ) ) )  e.  U )
6562, 17, 27wunstr 13447 . . . . . . . . . . . . . 14  |-  ( ph  ->  (comp `  X )  e.  U )
6617, 65wunrn 8564 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  (comp `  X
)  e.  U )
6717, 66wununi 8541 . . . . . . . . . . . 12  |-  ( ph  ->  U. ran  (comp `  X )  e.  U
)
6817, 67wunrn 8564 . . . . . . . . . . 11  |-  ( ph  ->  ran  U. ran  (comp `  X )  e.  U
)
6917, 68wununi 8541 . . . . . . . . . 10  |-  ( ph  ->  U. ran  U. ran  (comp `  X )  e.  U )
7017, 69wunpw 8542 . . . . . . . . 9  |-  ( ph  ->  ~P U. ran  U. ran  (comp `  X )  e.  U )
7162, 17, 30wunstr 13447 . . . . . . . . . . . . . 14  |-  ( ph  ->  (comp `  Y )  e.  U )
7217, 71wunrn 8564 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  (comp `  Y
)  e.  U )
7317, 72wununi 8541 . . . . . . . . . . . 12  |-  ( ph  ->  U. ran  (comp `  Y )  e.  U
)
7417, 73wunrn 8564 . . . . . . . . . . 11  |-  ( ph  ->  ran  U. ran  (comp `  Y )  e.  U
)
7517, 74wununi 8541 . . . . . . . . . 10  |-  ( ph  ->  U. ran  U. ran  (comp `  Y )  e.  U )
7617, 75wunpw 8542 . . . . . . . . 9  |-  ( ph  ->  ~P U. ran  U. ran  (comp `  Y )  e.  U )
7717, 70, 76wunxp 8559 . . . . . . . 8  |-  ( ph  ->  ( ~P U. ran  U.
ran  (comp `  X )  X.  ~P U. ran  U. ran  (comp `  Y )
)  e.  U )
7817, 60wunrn 8564 . . . . . . . . . 10  |-  ( ph  ->  ran  (  Hom  `  T
)  e.  U )
7917, 78wununi 8541 . . . . . . . . 9  |-  ( ph  ->  U. ran  (  Hom  `  T )  e.  U
)
8017, 79, 79wunxp 8559 . . . . . . . 8  |-  ( ph  ->  ( U. ran  (  Hom  `  T )  X. 
U. ran  (  Hom  `  T ) )  e.  U )
8117, 77, 80wunpm 8560 . . . . . . 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 5705 . . . . . . . . . . . . . . . . 17  |-  (comp `  X )  e.  _V
8382rnex 5096 . . . . . . . . . . . . . . . 16  |-  ran  (comp `  X )  e.  _V
8483uniex 4668 . . . . . . . . . . . . . . 15  |-  U. ran  (comp `  X )  e. 
_V
8584rnex 5096 . . . . . . . . . . . . . 14  |-  ran  U. ran  (comp `  X )  e.  _V
8685uniex 4668 . . . . . . . . . . . . 13  |-  U. ran  U.
ran  (comp `  X )  e.  _V
8786pwex 4346 . . . . . . . . . . . 12  |-  ~P U. ran  U. ran  (comp `  X )  e.  _V
88 fvex 5705 . . . . . . . . . . . . . . . . 17  |-  (comp `  Y )  e.  _V
8988rnex 5096 . . . . . . . . . . . . . . . 16  |-  ran  (comp `  Y )  e.  _V
9089uniex 4668 . . . . . . . . . . . . . . 15  |-  U. ran  (comp `  Y )  e. 
_V
9190rnex 5096 . . . . . . . . . . . . . 14  |-  ran  U. ran  (comp `  Y )  e.  _V
9291uniex 4668 . . . . . . . . . . . . 13  |-  U. ran  U.
ran  (comp `  Y )  e.  _V
9392pwex 4346 . . . . . . . . . . . 12  |-  ~P U. ran  U. ran  (comp `  Y )  e.  _V
9487, 93xpex 4953 . . . . . . . . . . 11  |-  ( ~P
U. ran  U. ran  (comp `  X )  X.  ~P U.
ran  U. ran  (comp `  Y ) )  e. 
_V
95 fvex 5705 . . . . . . . . . . . . . 14  |-  (  Hom  `  T )  e.  _V
9695rnex 5096 . . . . . . . . . . . . 13  |-  ran  (  Hom  `  T )  e. 
_V
9796uniex 4668 . . . . . . . . . . . 12  |-  U. ran  (  Hom  `  T )  e.  _V
9897, 97xpex 4953 . . . . . . . . . . 11  |-  ( U. ran  (  Hom  `  T
)  X.  U. ran  (  Hom  `  T )
)  e.  _V
99 ovssunirn 6070 . . . . . . . . . . . . . . . 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 6070 . . . . . . . . . . . . . . . . 17  |-  ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  X )
( 1st `  y
) )  C_  U. ran  (comp `  X )
101 rnss 5061 . . . . . . . . . . . . . . . . 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 4000 . . . . . . . . . . . . . . . . 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 3321 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  g ) ( <. ( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  X )
( 1st `  y
) ) ( 1st `  f ) )  C_  U.
ran  U. ran  (comp `  X )
105 ovex 6069 . . . . . . . . . . . . . . . 16  |-  ( ( 1st `  g ) ( <. ( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>. (comp `  X )
( 1st `  y
) ) ( 1st `  f ) )  e. 
_V
106105elpw 3769 . . . . . . . . . . . . . . 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 201 . . . . . . . . . . . . . 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 6070 . . . . . . . . . . . . . . . 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 6070 . . . . . . . . . . . . . . . . 17  |-  ( <.
( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>. (comp `  Y )
( 2nd `  y
) )  C_  U. ran  (comp `  Y )
110 rnss 5061 . . . . . . . . . . . . . . . . 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 4000 . . . . . . . . . . . . . . . . 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 3321 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  g ) ( <. ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>. (comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) )  C_  U.
ran  U. ran  (comp `  Y )
114 ovex 6069 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  g ) ( <. ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>. (comp `  Y )
( 2nd `  y
) ) ( 2nd `  f ) )  e. 
_V
115114elpw 3769 . . . . . . . . . . . . . . 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 201 . . . . . . . . . . . . . 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 4873 . . . . . . . . . . . . . 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 654 . . . . . . . . . . . . 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 2738 . . . . . . . . . . . 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 2408 . . . . . . . . . . . . 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 6381 . . . . . . . . . . . 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 200 . . . . . . . . . . 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 6070 . . . . . . . . . . . 12  |-  ( ( 2nd `  x ) (  Hom  `  T
) y )  C_  U.
ran  (  Hom  `  T
)
124 fvssunirn 5717 . . . . . . . . . . . 12  |-  ( (  Hom  `  T ) `  x )  C_  U. ran  (  Hom  `  T )
125 xpss12 4944 . . . . . . . . . . . 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 654 . . . . . . . . . . 11  |-  ( ( ( 2nd `  x
) (  Hom  `  T
) y )  X.  ( (  Hom  `  T
) `  x )
)  C_  ( U. ran  (  Hom  `  T
)  X.  U. ran  (  Hom  `  T )
)
127 elpm2r 6997 . . . . . . . . . . 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 655 . . . . . . . . . 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 2738 . . . . . . . . 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 2408 . . . . . . . . . 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 6381 . . . . . . . . 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 200 . . . . . . . 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 8562 . . . . . 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 8557 . . . . 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 8546 . . . 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 2482 . . 3  |-  ( ph  ->  T  e.  U )
138 inss2 3526 . . . . 5  |-  ( U  i^i  Cat )  C_  Cat
139138, 26sseldi 3310 . . . 4  |-  ( ph  ->  X  e.  Cat )
140138, 29sseldi 3310 . . . 4  |-  ( ph  ->  Y  e.  Cat )
1411, 139, 140xpccat 14246 . . 3  |-  ( ph  ->  T  e.  Cat )
142 elin 3494 . . 3  |-  ( T  e.  ( U  i^i  Cat )  <->  ( T  e.  U  /\  T  e. 
Cat ) )
143137, 141, 142sylanbrc 646 . 2  |-  ( ph  ->  T  e.  ( U  i^i  Cat ) )
144143, 25eleqtrrd 2485 1  |-  ( ph  ->  T  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   A.wral 2670   _Vcvv 2920    i^i cin 3283    C_ wss 3284   ~Pcpw 3763   {ctp 3780   <.cop 3781   U.cuni 3979   omcom 4808    X. cxp 4839   ran crn 4842   -->wf 5413   ` cfv 5417  (class class class)co 6044    e. cmpt2 6046   1stc1st 6310   2ndc2nd 6311    ^pm cpm 6982  WUnicwun 8535   1c1 8951   4c4 10011   5c5 10012  ;cdc 10342   ndxcnx 13425   Basecbs 13428    Hom chom 13499  compcco 13500   Catccat 13848  CatCatccatc 14208    X.c cxpc 14224
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-omul 6692  df-er 6868  df-ec 6870  df-qs 6874  df-map 6983  df-pm 6984  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-wun 8537  df-ni 8709  df-pli 8710  df-mi 8711  df-lti 8712  df-plpq 8745  df-mpq 8746  df-ltpq 8747  df-enq 8748  df-nq 8749  df-erq 8750  df-plq 8751  df-mq 8752  df-1nq 8753  df-rq 8754  df-ltnq 8755  df-np 8818  df-plp 8820  df-ltp 8822  df-enr 8894  df-nr 8895  df-c 8956  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-5 10021  df-6 10022  df-7 10023  df-8 10024  df-9 10025  df-10 10026  df-n0 10182  df-z 10243  df-dec 10343  df-uz 10449  df-fz 11004  df-struct 13430  df-ndx 13431  df-slot 13432  df-base 13433  df-hom 13512  df-cco 13513  df-cat 13852  df-cid 13853  df-catc 14209  df-xpc 14228
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