MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xkoval Structured version   Unicode version

Theorem xkoval 20526
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 462 . . . . . . . . . . . . 13  |-  ( ( s  =  S  /\  r  =  R )  ->  r  =  R )
21unieqd 4223 . . . . . . . . . . . 12  |-  ( ( s  =  S  /\  r  =  R )  ->  U. r  =  U. R )
3 xkoval.x . . . . . . . . . . . 12  |-  X  = 
U. R
42, 3syl6eqr 2479 . . . . . . . . . . 11  |-  ( ( s  =  S  /\  r  =  R )  ->  U. r  =  X )
54pweqd 3981 . . . . . . . . . 10  |-  ( ( s  =  S  /\  r  =  R )  ->  ~P U. r  =  ~P X )
61oveq1d 6311 . . . . . . . . . . 11  |-  ( ( s  =  S  /\  r  =  R )  ->  ( rt  x )  =  ( Rt  x ) )
76eleq1d 2489 . . . . . . . . . 10  |-  ( ( s  =  S  /\  r  =  R )  ->  ( ( rt  x )  e.  Comp  <->  ( Rt  x )  e.  Comp ) )
85, 7rabeqbidv 3073 . . . . . . . . 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 2479 . . . . . . . 8  |-  ( ( s  =  S  /\  r  =  R )  ->  { x  e.  ~P U. r  |  ( rt  x )  e.  Comp }  =  K )
11 simpl 458 . . . . . . . 8  |-  ( ( s  =  S  /\  r  =  R )  ->  s  =  S )
121, 11oveq12d 6314 . . . . . . . . 9  |-  ( ( s  =  S  /\  r  =  R )  ->  ( r  Cn  s
)  =  ( R  Cn  S ) )
13 rabeq 3072 . . . . . . . . 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 17 . . . . . . . 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 6358 . . . . . . 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 2479 . . . . . 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 5073 . . . . 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 5876 . . . 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 5876 . . 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 20502 . . 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 5882 . . 3  |-  ( topGen `  ( fi `  ran  T ) )  e.  _V
2320, 21, 22ovmpt2a 6432 . 2  |-  ( ( S  e.  Top  /\  R  e.  Top )  ->  ( S  ^ko  R )  =  (
topGen `  ( fi `  ran  T ) ) )
2423ancoms 454 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 370    = wceq 1437    e. wcel 1867   {crab 2777    C_ wss 3433   ~Pcpw 3976   U.cuni 4213   ran crn 4846   "cima 4848   ` cfv 5592  (class class class)co 6296    |-> cmpt2 6298   ficfi 7921   ↾t crest 15271   topGenctg 15288   Topctop 19841    Cn ccn 20164   Compccmp 20325    ^ko cxko 20500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5556  df-fun 5594  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-xko 20502
This theorem is referenced by:  xkotop  20527  xkoopn  20528  xkouni  20538  xkoccn  20558  xkopt  20594  xkoco1cn  20596  xkoco2cn  20597  xkococn  20599  xkoinjcn  20626
  Copyright terms: Public domain W3C validator