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Theorem xkoval 19851
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 4255 . . . . . . . . . . . 12  |-  ( ( s  =  S  /\  r  =  R )  ->  U. r  =  U. R )
3 xkoval.x . . . . . . . . . . . 12  |-  X  = 
U. R
42, 3syl6eqr 2526 . . . . . . . . . . 11  |-  ( ( s  =  S  /\  r  =  R )  ->  U. r  =  X )
54pweqd 4015 . . . . . . . . . 10  |-  ( ( s  =  S  /\  r  =  R )  ->  ~P U. r  =  ~P X )
61oveq1d 6299 . . . . . . . . . . 11  |-  ( ( s  =  S  /\  r  =  R )  ->  ( rt  x )  =  ( Rt  x ) )
76eleq1d 2536 . . . . . . . . . 10  |-  ( ( s  =  S  /\  r  =  R )  ->  ( ( rt  x )  e.  Comp  <->  ( Rt  x )  e.  Comp ) )
85, 7rabeqbidv 3108 . . . . . . . . 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 2526 . . . . . . . 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 6302 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  ( r  Cn  s
)  =  ( R  Cn  S ) )
13 rabeq 3107 . . . . . . . . 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 6343 . . . . . . 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 2526 . . . . . 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 5230 . . . . 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 5870 . . . 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 5870 . . 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 19827 . . 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 5876 . . 3  |-  ( topGen `  ( fi `  ran  T ) )  e.  _V
2320, 21, 22ovmpt2a 6417 . 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 1379    e. wcel 1767   {crab 2818    C_ wss 3476   ~Pcpw 4010   U.cuni 4245   ran crn 5000   "cima 5002   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   ficfi 7870   ↾t crest 14676   topGenctg 14693   Topctop 19189    Cn ccn 19519   Compccmp 19680    ^ko cxko 19825
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  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-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-xko 19827
This theorem is referenced by:  xkotop  19852  xkoopn  19853  xkouni  19863  xkoccn  19883  xkopt  19919  xkoco1cn  19921  xkoco2cn  19922  xkococn  19924  xkoinjcn  19951
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