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Theorem xkoopn 19280
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 6217 . . . . . . 7  |-  ( R  Cn  S )  e. 
_V
21pwex 4575 . . . . . 6  |-  ~P ( R  Cn  S )  e. 
_V
3 xkoopn.x . . . . . . . 8  |-  X  = 
U. R
4 eqid 2451 . . . . . . . 8  |-  { x  e.  ~P X  |  ( Rt  x )  e.  Comp }  =  { x  e. 
~P X  |  ( Rt  x )  e.  Comp }
5 eqid 2451 . . . . . . . 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 19276 . . . . . . 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 5665 . . . . . . 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 4537 . . . . 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 7772 . . . . 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 5801 . . . . 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 18689 . . . . 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 3465 . . 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 18637 . . . . . . . 8  |-  ( R  e.  Top  ->  X  e.  R )
19 elpw2g 4555 . . . . . . . 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 6200 . . . . . . . 8  |-  ( x  =  A  ->  ( Rt  x )  =  ( Rt  A ) )
2423eleq1d 2520 . . . . . . 7  |-  ( x  =  A  ->  (
( Rt  x )  e.  Comp  <->  ( Rt  A )  e.  Comp ) )
2524elrab 3216 . . . . . 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 2452 . . . . 5  |-  ( ph  ->  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  U }  =  { f  e.  ( R  Cn  S
)  |  ( f
" A )  C_  U } )
29 imaeq2 5265 . . . . . . . . 9  |-  ( k  =  A  ->  (
f " k )  =  ( f " A ) )
3029sseq1d 3483 . . . . . . . 8  |-  ( k  =  A  ->  (
( f " k
)  C_  v  <->  ( f " A )  C_  v
) )
3130rabbidv 3062 . . . . . . 7  |-  ( k  =  A  ->  { f  e.  ( R  Cn  S )  |  ( f " k ) 
C_  v }  =  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  v } )
3231eqeq2d 2465 . . . . . 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 3478 . . . . . . . 8  |-  ( v  =  U  ->  (
( f " A
)  C_  v  <->  ( f " A )  C_  U
) )
3433rabbidv 3062 . . . . . . 7  |-  ( v  =  U  ->  { f  e.  ( R  Cn  S )  |  ( f " A ) 
C_  v }  =  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  U } )
3534eqeq2d 2465 . . . . . 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 3180 . . . . 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 1219 . . . 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 4543 . . . . 5  |-  { f  e.  ( R  Cn  S )  |  ( f " A ) 
C_  U }  e.  _V
39 eqeq1 2455 . . . . . 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 2867 . . . . 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 6302 . . . . 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 3208 . . . 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 3454 . 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 19278 . . 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 2542 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 1370    e. wcel 1758   E.wrex 2796   {crab 2799   _Vcvv 3070    C_ wss 3428   ~Pcpw 3960   U.cuni 4191    X. cxp 4938   ran crn 4941   "cima 4943   -->wf 5514   ` cfv 5518  (class class class)co 6192    |-> cmpt2 6194   ficfi 7763   ↾t crest 14463   topGenctg 14480   Topctop 18616    Cn ccn 18946   Compccmp 19107    ^ko cxko 19252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-1o 7022  df-en 7413  df-fin 7416  df-fi 7764  df-topgen 14486  df-top 18621  df-xko 19254
This theorem is referenced by:  xkouni  19290  xkohaus  19344  xkoptsub  19345  xkoco1cn  19348  xkoco2cn  19349  xkococnlem  19350
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