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Theorem xkoval 19158
Description: Value of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
Hypotheses
Ref Expression
xkoval.x  |-  X  = 
U. R
xkoval.k  |-  K  =  { x  e.  ~P X  |  ( Rt  x
)  e.  Comp }
xkoval.t  |-  T  =  ( k  e.  K ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } )
Assertion
Ref Expression
xkoval  |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( S  ^ko  R )  =  (
topGen `  ( fi `  ran  T ) ) )
Distinct variable groups:    v, k, K    f, k, v, x, R    S, f, k, v, x    k, X, x
Allowed substitution hints:    T( x, v, f, k)    K( x, f)    X( v, f)

Proof of Theorem xkoval
Dummy variables  s 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 461 . . . . . . . . . . . . 13  |-  ( ( s  =  S  /\  r  =  R )  ->  r  =  R )
21unieqd 4099 . . . . . . . . . . . 12  |-  ( ( s  =  S  /\  r  =  R )  ->  U. r  =  U. R )
3 xkoval.x . . . . . . . . . . . 12  |-  X  = 
U. R
42, 3syl6eqr 2491 . . . . . . . . . . 11  |-  ( ( s  =  S  /\  r  =  R )  ->  U. r  =  X )
54pweqd 3863 . . . . . . . . . 10  |-  ( ( s  =  S  /\  r  =  R )  ->  ~P U. r  =  ~P X )
61oveq1d 6104 . . . . . . . . . . 11  |-  ( ( s  =  S  /\  r  =  R )  ->  ( rt  x )  =  ( Rt  x ) )
76eleq1d 2507 . . . . . . . . . 10  |-  ( ( s  =  S  /\  r  =  R )  ->  ( ( rt  x )  e.  Comp  <->  ( Rt  x )  e.  Comp ) )
85, 7rabeqbidv 2965 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  { x  e.  ~P U. r  |  ( rt  x )  e.  Comp }  =  { x  e.  ~P X  |  ( Rt  x
)  e.  Comp } )
9 xkoval.k . . . . . . . . 9  |-  K  =  { x  e.  ~P X  |  ( Rt  x
)  e.  Comp }
108, 9syl6eqr 2491 . . . . . . . 8  |-  ( ( s  =  S  /\  r  =  R )  ->  { x  e.  ~P U. r  |  ( rt  x )  e.  Comp }  =  K )
11 simpl 457 . . . . . . . 8  |-  ( ( s  =  S  /\  r  =  R )  ->  s  =  S )
121, 11oveq12d 6107 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  ( r  Cn  s
)  =  ( R  Cn  S ) )
13 rabeq 2964 . . . . . . . . 9  |-  ( ( r  Cn  s )  =  ( R  Cn  S )  ->  { f  e.  ( r  Cn  s )  |  ( f " k ) 
C_  v }  =  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } )
1412, 13syl 16 . . . . . . . 8  |-  ( ( s  =  S  /\  r  =  R )  ->  { f  e.  ( r  Cn  s )  |  ( f "
k )  C_  v }  =  { f  e.  ( R  Cn  S
)  |  ( f
" k )  C_  v } )
1510, 11, 14mpt2eq123dv 6146 . . . . . . 7  |-  ( ( s  =  S  /\  r  =  R )  ->  ( k  e.  {
x  e.  ~P U. r  |  ( rt  x
)  e.  Comp } , 
v  e.  s  |->  { f  e.  ( r  Cn  s )  |  ( f " k
)  C_  v }
)  =  ( k  e.  K ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f " k ) 
C_  v } ) )
16 xkoval.t . . . . . . 7  |-  T  =  ( k  e.  K ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } )
1715, 16syl6eqr 2491 . . . . . 6  |-  ( ( s  =  S  /\  r  =  R )  ->  ( k  e.  {
x  e.  ~P U. r  |  ( rt  x
)  e.  Comp } , 
v  e.  s  |->  { f  e.  ( r  Cn  s )  |  ( f " k
)  C_  v }
)  =  T )
1817rneqd 5065 . . . . 5  |-  ( ( s  =  S  /\  r  =  R )  ->  ran  ( k  e. 
{ x  e.  ~P U. r  |  ( rt  x )  e.  Comp } , 
v  e.  s  |->  { f  e.  ( r  Cn  s )  |  ( f " k
)  C_  v }
)  =  ran  T
)
1918fveq2d 5693 . . . 4  |-  ( ( s  =  S  /\  r  =  R )  ->  ( fi `  ran  ( k  e.  {
x  e.  ~P U. r  |  ( rt  x
)  e.  Comp } , 
v  e.  s  |->  { f  e.  ( r  Cn  s )  |  ( f " k
)  C_  v }
) )  =  ( fi `  ran  T
) )
2019fveq2d 5693 . . 3  |-  ( ( s  =  S  /\  r  =  R )  ->  ( topGen `  ( fi ` 
ran  ( k  e. 
{ x  e.  ~P U. r  |  ( rt  x )  e.  Comp } , 
v  e.  s  |->  { f  e.  ( r  Cn  s )  |  ( f " k
)  C_  v }
) ) )  =  ( topGen `  ( fi ` 
ran  T ) ) )
21 df-xko 19134 . . 3  |-  ^ko  =  ( s  e.  Top ,  r  e. 
Top  |->  ( topGen `  ( fi `  ran  ( k  e.  { x  e. 
~P U. r  |  ( rt  x )  e.  Comp } ,  v  e.  s 
|->  { f  e.  ( r  Cn  s )  |  ( f "
k )  C_  v } ) ) ) )
22 fvex 5699 . . 3  |-  ( topGen `  ( fi `  ran  T ) )  e.  _V
2320, 21, 22ovmpt2a 6219 . 2  |-  ( ( S  e.  Top  /\  R  e.  Top )  ->  ( S  ^ko  R )  =  (
topGen `  ( fi `  ran  T ) ) )
2423ancoms 453 1  |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( S  ^ko  R )  =  (
topGen `  ( fi `  ran  T ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2717    C_ wss 3326   ~Pcpw 3858   U.cuni 4089   ran crn 4839   "cima 4841   ` cfv 5416  (class class class)co 6089    e. cmpt2 6091   ficfi 7658   ↾t crest 14357   topGenctg 14374   Topctop 18496    Cn ccn 18826   Compccmp 18987    ^ko cxko 19132
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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-iota 5379  df-fun 5418  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-xko 19134
This theorem is referenced by:  xkotop  19159  xkoopn  19160  xkouni  19170  xkoccn  19190  xkopt  19226  xkoco1cn  19228  xkoco2cn  19229  xkococn  19231  xkoinjcn  19258
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