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Theorem xkoopn 20653
Description: A basic open set of the compact-open topology. (Contributed by Mario Carneiro, 19-Mar-2015.)
Hypotheses
Ref Expression
xkoopn.x  |-  X  = 
U. R
xkoopn.r  |-  ( ph  ->  R  e.  Top )
xkoopn.s  |-  ( ph  ->  S  e.  Top )
xkoopn.a  |-  ( ph  ->  A  C_  X )
xkoopn.c  |-  ( ph  ->  ( Rt  A )  e.  Comp )
xkoopn.u  |-  ( ph  ->  U  e.  S )
Assertion
Ref Expression
xkoopn  |-  ( ph  ->  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  U }  e.  ( S  ^ko  R ) )
Distinct variable groups:    A, f    R, f    S, f    U, f
Allowed substitution hints:    ph( f)    X( f)

Proof of Theorem xkoopn
Dummy variables  k 
v  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6343 . . . . . . 7  |-  ( R  Cn  S )  e. 
_V
21pwex 4600 . . . . . 6  |-  ~P ( R  Cn  S )  e. 
_V
3 xkoopn.x . . . . . . . 8  |-  X  = 
U. R
4 eqid 2462 . . . . . . . 8  |-  { x  e.  ~P X  |  ( Rt  x )  e.  Comp }  =  { x  e. 
~P X  |  ( Rt  x )  e.  Comp }
5 eqid 2462 . . . . . . . 8  |-  ( k  e.  { x  e. 
~P X  |  ( Rt  x )  e.  Comp } ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } )  =  ( k  e.  { x  e.  ~P X  |  ( Rt  x )  e.  Comp } ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } )
63, 4, 5xkotf 20649 . . . . . . 7  |-  ( k  e.  { x  e. 
~P X  |  ( Rt  x )  e.  Comp } ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } ) : ( { x  e.  ~P X  |  ( Rt  x
)  e.  Comp }  X.  S ) --> ~P ( R  Cn  S )
7 frn 5758 . . . . . . 7  |-  ( ( k  e.  { x  e.  ~P X  |  ( Rt  x )  e.  Comp } ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } ) : ( { x  e.  ~P X  |  ( Rt  x
)  e.  Comp }  X.  S ) --> ~P ( R  Cn  S )  ->  ran  ( k  e.  {
x  e.  ~P X  |  ( Rt  x )  e.  Comp } ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f " k ) 
C_  v } ) 
C_  ~P ( R  Cn  S ) )
86, 7ax-mp 5 . . . . . 6  |-  ran  (
k  e.  { x  e.  ~P X  |  ( Rt  x )  e.  Comp } ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } )  C_  ~P ( R  Cn  S
)
92, 8ssexi 4562 . . . . 5  |-  ran  (
k  e.  { x  e.  ~P X  |  ( Rt  x )  e.  Comp } ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } )  e.  _V
10 ssfii 7959 . . . . 5  |-  ( ran  ( k  e.  {
x  e.  ~P X  |  ( Rt  x )  e.  Comp } ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f " k ) 
C_  v } )  e.  _V  ->  ran  ( k  e.  {
x  e.  ~P X  |  ( Rt  x )  e.  Comp } ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f " k ) 
C_  v } ) 
C_  ( fi `  ran  ( k  e.  {
x  e.  ~P X  |  ( Rt  x )  e.  Comp } ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f " k ) 
C_  v } ) ) )
119, 10ax-mp 5 . . . 4  |-  ran  (
k  e.  { x  e.  ~P X  |  ( Rt  x )  e.  Comp } ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } )  C_  ( fi `  ran  ( k  e.  { x  e. 
~P X  |  ( Rt  x )  e.  Comp } ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } ) )
12 fvex 5898 . . . . 5  |-  ( fi
`  ran  ( k  e.  { x  e.  ~P X  |  ( Rt  x
)  e.  Comp } , 
v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f " k
)  C_  v }
) )  e.  _V
13 bastg 20030 . . . . 5  |-  ( ( fi `  ran  (
k  e.  { x  e.  ~P X  |  ( Rt  x )  e.  Comp } ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } ) )  e. 
_V  ->  ( fi `  ran  ( k  e.  {
x  e.  ~P X  |  ( Rt  x )  e.  Comp } ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f " k ) 
C_  v } ) )  C_  ( topGen `  ( fi `  ran  ( k  e.  {
x  e.  ~P X  |  ( Rt  x )  e.  Comp } ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f " k ) 
C_  v } ) ) ) )
1412, 13ax-mp 5 . . . 4  |-  ( fi
`  ran  ( k  e.  { x  e.  ~P X  |  ( Rt  x
)  e.  Comp } , 
v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f " k
)  C_  v }
) )  C_  ( topGen `
 ( fi `  ran  ( k  e.  {
x  e.  ~P X  |  ( Rt  x )  e.  Comp } ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f " k ) 
C_  v } ) ) )
1511, 14sstri 3453 . . 3  |-  ran  (
k  e.  { x  e.  ~P X  |  ( Rt  x )  e.  Comp } ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } )  C_  ( topGen `
 ( fi `  ran  ( k  e.  {
x  e.  ~P X  |  ( Rt  x )  e.  Comp } ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f " k ) 
C_  v } ) ) )
16 xkoopn.a . . . . . . 7  |-  ( ph  ->  A  C_  X )
17 xkoopn.r . . . . . . . 8  |-  ( ph  ->  R  e.  Top )
183topopn 19985 . . . . . . . 8  |-  ( R  e.  Top  ->  X  e.  R )
19 elpw2g 4580 . . . . . . . 8  |-  ( X  e.  R  ->  ( A  e.  ~P X  <->  A 
C_  X ) )
2017, 18, 193syl 18 . . . . . . 7  |-  ( ph  ->  ( A  e.  ~P X 
<->  A  C_  X )
)
2116, 20mpbird 240 . . . . . 6  |-  ( ph  ->  A  e.  ~P X
)
22 xkoopn.c . . . . . 6  |-  ( ph  ->  ( Rt  A )  e.  Comp )
23 oveq2 6323 . . . . . . . 8  |-  ( x  =  A  ->  ( Rt  x )  =  ( Rt  A ) )
2423eleq1d 2524 . . . . . . 7  |-  ( x  =  A  ->  (
( Rt  x )  e.  Comp  <->  ( Rt  A )  e.  Comp ) )
2524elrab 3208 . . . . . 6  |-  ( A  e.  { x  e. 
~P X  |  ( Rt  x )  e.  Comp }  <-> 
( A  e.  ~P X  /\  ( Rt  A )  e.  Comp ) )
2621, 22, 25sylanbrc 675 . . . . 5  |-  ( ph  ->  A  e.  { x  e.  ~P X  |  ( Rt  x )  e.  Comp } )
27 xkoopn.u . . . . 5  |-  ( ph  ->  U  e.  S )
28 eqidd 2463 . . . . 5  |-  ( ph  ->  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  U }  =  { f  e.  ( R  Cn  S
)  |  ( f
" A )  C_  U } )
29 imaeq2 5183 . . . . . . . . 9  |-  ( k  =  A  ->  (
f " k )  =  ( f " A ) )
3029sseq1d 3471 . . . . . . . 8  |-  ( k  =  A  ->  (
( f " k
)  C_  v  <->  ( f " A )  C_  v
) )
3130rabbidv 3048 . . . . . . 7  |-  ( k  =  A  ->  { f  e.  ( R  Cn  S )  |  ( f " k ) 
C_  v }  =  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  v } )
3231eqeq2d 2472 . . . . . 6  |-  ( k  =  A  ->  ( { f  e.  ( R  Cn  S )  |  ( f " A )  C_  U }  =  { f  e.  ( R  Cn  S
)  |  ( f
" k )  C_  v }  <->  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  U }  =  { f  e.  ( R  Cn  S
)  |  ( f
" A )  C_  v } ) )
33 sseq2 3466 . . . . . . . 8  |-  ( v  =  U  ->  (
( f " A
)  C_  v  <->  ( f " A )  C_  U
) )
3433rabbidv 3048 . . . . . . 7  |-  ( v  =  U  ->  { f  e.  ( R  Cn  S )  |  ( f " A ) 
C_  v }  =  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  U } )
3534eqeq2d 2472 . . . . . 6  |-  ( v  =  U  ->  ( { f  e.  ( R  Cn  S )  |  ( f " A )  C_  U }  =  { f  e.  ( R  Cn  S
)  |  ( f
" A )  C_  v }  <->  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  U }  =  { f  e.  ( R  Cn  S
)  |  ( f
" A )  C_  U } ) )
3632, 35rspc2ev 3173 . . . . 5  |-  ( ( A  e.  { x  e.  ~P X  |  ( Rt  x )  e.  Comp }  /\  U  e.  S  /\  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  U }  =  { f  e.  ( R  Cn  S
)  |  ( f
" A )  C_  U } )  ->  E. k  e.  { x  e.  ~P X  |  ( Rt  x
)  e.  Comp } E. v  e.  S  {
f  e.  ( R  Cn  S )  |  ( f " A
)  C_  U }  =  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } )
3726, 27, 28, 36syl3anc 1276 . . . 4  |-  ( ph  ->  E. k  e.  {
x  e.  ~P X  |  ( Rt  x )  e.  Comp } E. v  e.  S  { f  e.  ( R  Cn  S
)  |  ( f
" A )  C_  U }  =  {
f  e.  ( R  Cn  S )  |  ( f " k
)  C_  v }
)
381rabex 4568 . . . . 5  |-  { f  e.  ( R  Cn  S )  |  ( f " A ) 
C_  U }  e.  _V
39 eqeq1 2466 . . . . . 6  |-  ( y  =  { f  e.  ( R  Cn  S
)  |  ( f
" A )  C_  U }  ->  ( y  =  { f  e.  ( R  Cn  S
)  |  ( f
" k )  C_  v }  <->  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  U }  =  { f  e.  ( R  Cn  S
)  |  ( f
" k )  C_  v } ) )
40392rexbidv 2920 . . . . 5  |-  ( y  =  { f  e.  ( R  Cn  S
)  |  ( f
" A )  C_  U }  ->  ( E. k  e.  { x  e.  ~P X  |  ( Rt  x )  e.  Comp } E. v  e.  S  y  =  { f  e.  ( R  Cn  S
)  |  ( f
" k )  C_  v }  <->  E. k  e.  {
x  e.  ~P X  |  ( Rt  x )  e.  Comp } E. v  e.  S  { f  e.  ( R  Cn  S
)  |  ( f
" A )  C_  U }  =  {
f  e.  ( R  Cn  S )  |  ( f " k
)  C_  v }
) )
415rnmpt2 6433 . . . . 5  |-  ran  (
k  e.  { x  e.  ~P X  |  ( Rt  x )  e.  Comp } ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } )  =  {
y  |  E. k  e.  { x  e.  ~P X  |  ( Rt  x
)  e.  Comp } E. v  e.  S  y  =  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } }
4238, 40, 41elab2 3200 . . . 4  |-  ( { f  e.  ( R  Cn  S )  |  ( f " A
)  C_  U }  e.  ran  ( k  e. 
{ x  e.  ~P X  |  ( Rt  x
)  e.  Comp } , 
v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f " k
)  C_  v }
)  <->  E. k  e.  {
x  e.  ~P X  |  ( Rt  x )  e.  Comp } E. v  e.  S  { f  e.  ( R  Cn  S
)  |  ( f
" A )  C_  U }  =  {
f  e.  ( R  Cn  S )  |  ( f " k
)  C_  v }
)
4337, 42sylibr 217 . . 3  |-  ( ph  ->  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  U }  e.  ran  ( k  e.  { x  e. 
~P X  |  ( Rt  x )  e.  Comp } ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f "
k )  C_  v } ) )
4415, 43sseldi 3442 . 2  |-  ( ph  ->  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  U }  e.  ( topGen `  ( fi `  ran  ( k  e.  {
x  e.  ~P X  |  ( Rt  x )  e.  Comp } ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f " k ) 
C_  v } ) ) ) )
45 xkoopn.s . . 3  |-  ( ph  ->  S  e.  Top )
463, 4, 5xkoval 20651 . . 3  |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( S  ^ko  R )  =  (
topGen `  ( fi `  ran  ( k  e.  {
x  e.  ~P X  |  ( Rt  x )  e.  Comp } ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f " k ) 
C_  v } ) ) ) )
4717, 45, 46syl2anc 671 . 2  |-  ( ph  ->  ( S  ^ko  R )  =  (
topGen `  ( fi `  ran  ( k  e.  {
x  e.  ~P X  |  ( Rt  x )  e.  Comp } ,  v  e.  S  |->  { f  e.  ( R  Cn  S )  |  ( f " k ) 
C_  v } ) ) ) )
4844, 47eleqtrrd 2543 1  |-  ( ph  ->  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  U }  e.  ( S  ^ko  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1455    e. wcel 1898   E.wrex 2750   {crab 2753   _Vcvv 3057    C_ wss 3416   ~Pcpw 3963   U.cuni 4212    X. cxp 4851   ran crn 4854   "cima 4856   -->wf 5597   ` cfv 5601  (class class class)co 6315    |-> cmpt2 6317   ficfi 7950   ↾t crest 15368   topGenctg 15385   Topctop 19966    Cn ccn 20289   Compccmp 20450    ^ko cxko 20625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-1st 6820  df-2nd 6821  df-1o 7208  df-en 7596  df-fin 7599  df-fi 7951  df-topgen 15391  df-top 19970  df-xko 20627
This theorem is referenced by:  xkouni  20663  xkohaus  20717  xkoptsub  20718  xkoco1cn  20721  xkoco2cn  20722  xkococnlem  20723
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