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Theorem xkoopn 19825
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 6307 . . . . . . 7  |-  ( R  Cn  S )  e. 
_V
21pwex 4630 . . . . . 6  |-  ~P ( R  Cn  S )  e. 
_V
3 xkoopn.x . . . . . . . 8  |-  X  = 
U. R
4 eqid 2467 . . . . . . . 8  |-  { x  e.  ~P X  |  ( Rt  x )  e.  Comp }  =  { x  e. 
~P X  |  ( Rt  x )  e.  Comp }
5 eqid 2467 . . . . . . . 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 19821 . . . . . . 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 5735 . . . . . . 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 4592 . . . . 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 7875 . . . . 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 5874 . . . . 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 19234 . . . . 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 3513 . . 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 19182 . . . . . . . 8  |-  ( R  e.  Top  ->  X  e.  R )
19 elpw2g 4610 . . . . . . . 8  |-  ( X  e.  R  ->  ( A  e.  ~P X  <->  A 
C_  X ) )
2017, 18, 193syl 20 . . . . . . 7  |-  ( ph  ->  ( A  e.  ~P X 
<->  A  C_  X )
)
2116, 20mpbird 232 . . . . . 6  |-  ( ph  ->  A  e.  ~P X
)
22 xkoopn.c . . . . . 6  |-  ( ph  ->  ( Rt  A )  e.  Comp )
23 oveq2 6290 . . . . . . . 8  |-  ( x  =  A  ->  ( Rt  x )  =  ( Rt  A ) )
2423eleq1d 2536 . . . . . . 7  |-  ( x  =  A  ->  (
( Rt  x )  e.  Comp  <->  ( Rt  A )  e.  Comp ) )
2524elrab 3261 . . . . . 6  |-  ( A  e.  { x  e. 
~P X  |  ( Rt  x )  e.  Comp }  <-> 
( A  e.  ~P X  /\  ( Rt  A )  e.  Comp ) )
2621, 22, 25sylanbrc 664 . . . . 5  |-  ( ph  ->  A  e.  { x  e.  ~P X  |  ( Rt  x )  e.  Comp } )
27 xkoopn.u . . . . 5  |-  ( ph  ->  U  e.  S )
28 eqidd 2468 . . . . 5  |-  ( ph  ->  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  U }  =  { f  e.  ( R  Cn  S
)  |  ( f
" A )  C_  U } )
29 imaeq2 5331 . . . . . . . . 9  |-  ( k  =  A  ->  (
f " k )  =  ( f " A ) )
3029sseq1d 3531 . . . . . . . 8  |-  ( k  =  A  ->  (
( f " k
)  C_  v  <->  ( f " A )  C_  v
) )
3130rabbidv 3105 . . . . . . 7  |-  ( k  =  A  ->  { f  e.  ( R  Cn  S )  |  ( f " k ) 
C_  v }  =  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  v } )
3231eqeq2d 2481 . . . . . 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 3526 . . . . . . . 8  |-  ( v  =  U  ->  (
( f " A
)  C_  v  <->  ( f " A )  C_  U
) )
3433rabbidv 3105 . . . . . . 7  |-  ( v  =  U  ->  { f  e.  ( R  Cn  S )  |  ( f " A ) 
C_  v }  =  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  U } )
3534eqeq2d 2481 . . . . . 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 3225 . . . . 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 1228 . . . 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 4598 . . . . 5  |-  { f  e.  ( R  Cn  S )  |  ( f " A ) 
C_  U }  e.  _V
39 eqeq1 2471 . . . . . 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 2980 . . . . 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 6394 . . . . 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 3253 . . . 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 212 . . 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 3502 . 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 19823 . . 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 661 . 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 2558 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 184    = wceq 1379    e. wcel 1767   E.wrex 2815   {crab 2818   _Vcvv 3113    C_ wss 3476   ~Pcpw 4010   U.cuni 4245    X. cxp 4997   ran crn 5000   "cima 5002   -->wf 5582   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   ficfi 7866   ↾t crest 14672   topGenctg 14689   Topctop 19161    Cn ccn 19491   Compccmp 19652    ^ko cxko 19797
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-8 1769  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-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-1o 7127  df-en 7514  df-fin 7517  df-fi 7867  df-topgen 14695  df-top 19166  df-xko 19799
This theorem is referenced by:  xkouni  19835  xkohaus  19889  xkoptsub  19890  xkoco1cn  19893  xkoco2cn  19894  xkococnlem  19895
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