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Theorem xkoopn 19142
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 6111 . . . . . . 7  |-  ( R  Cn  S )  e. 
_V
21pwex 4470 . . . . . 6  |-  ~P ( R  Cn  S )  e. 
_V
3 xkoopn.x . . . . . . . 8  |-  X  = 
U. R
4 eqid 2438 . . . . . . . 8  |-  { x  e.  ~P X  |  ( Rt  x )  e.  Comp }  =  { x  e. 
~P X  |  ( Rt  x )  e.  Comp }
5 eqid 2438 . . . . . . . 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 19138 . . . . . . 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 5560 . . . . . . 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 4432 . . . . 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 7661 . . . . 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 5696 . . . . 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 18551 . . . . 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 3360 . . 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 18499 . . . . . . . 8  |-  ( R  e.  Top  ->  X  e.  R )
19 elpw2g 4450 . . . . . . . 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 6094 . . . . . . . 8  |-  ( x  =  A  ->  ( Rt  x )  =  ( Rt  A ) )
2423eleq1d 2504 . . . . . . 7  |-  ( x  =  A  ->  (
( Rt  x )  e.  Comp  <->  ( Rt  A )  e.  Comp ) )
2524elrab 3112 . . . . . 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 2439 . . . . 5  |-  ( ph  ->  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  U }  =  { f  e.  ( R  Cn  S
)  |  ( f
" A )  C_  U } )
29 imaeq2 5160 . . . . . . . . 9  |-  ( k  =  A  ->  (
f " k )  =  ( f " A ) )
3029sseq1d 3378 . . . . . . . 8  |-  ( k  =  A  ->  (
( f " k
)  C_  v  <->  ( f " A )  C_  v
) )
3130rabbidv 2959 . . . . . . 7  |-  ( k  =  A  ->  { f  e.  ( R  Cn  S )  |  ( f " k ) 
C_  v }  =  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  v } )
3231eqeq2d 2449 . . . . . 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 3373 . . . . . . . 8  |-  ( v  =  U  ->  (
( f " A
)  C_  v  <->  ( f " A )  C_  U
) )
3433rabbidv 2959 . . . . . . 7  |-  ( v  =  U  ->  { f  e.  ( R  Cn  S )  |  ( f " A ) 
C_  v }  =  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  U } )
3534eqeq2d 2449 . . . . . 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 3076 . . . . 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 1218 . . . 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 4438 . . . . 5  |-  { f  e.  ( R  Cn  S )  |  ( f " A ) 
C_  U }  e.  _V
39 eqeq1 2444 . . . . . 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 2753 . . . . 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 6195 . . . . 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 3104 . . . 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 3349 . 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 19140 . . 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 2515 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 1369    e. wcel 1756   E.wrex 2711   {crab 2714   _Vcvv 2967    C_ wss 3323   ~Pcpw 3855   U.cuni 4086    X. cxp 4833   ran crn 4836   "cima 4838   -->wf 5409   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   ficfi 7652   ↾t crest 14351   topGenctg 14368   Topctop 18478    Cn ccn 18808   Compccmp 18969    ^ko cxko 19114
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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-1o 6912  df-en 7303  df-fin 7306  df-fi 7653  df-topgen 14374  df-top 18483  df-xko 19116
This theorem is referenced by:  xkouni  19152  xkohaus  19206  xkoptsub  19207  xkoco1cn  19210  xkoco2cn  19211  xkococnlem  19212
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