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Theorem xkoopn 17340
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  ^ k o  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 5925 . . . . . . 7  |-  ( R  Cn  S )  e. 
_V
21pwex 4230 . . . . . 6  |-  ~P ( R  Cn  S )  e. 
_V
3 xkoopn.x . . . . . . . 8  |-  X  = 
U. R
4 eqid 2316 . . . . . . . 8  |-  { x  e.  ~P X  |  ( Rt  x )  e.  Comp }  =  { x  e. 
~P X  |  ( Rt  x )  e.  Comp }
5 eqid 2316 . . . . . . . 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 17336 . . . . . . 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 5433 . . . . . . 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 8 . . . . . 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 4196 . . . . 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 7217 . . . . 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 8 . . . 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 5577 . . . . 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 16760 . . . . 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 8 . . . 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 3222 . . 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 16708 . . . . . . . 8  |-  ( R  e.  Top  ->  X  e.  R )
19 elpw2g 4211 . . . . . . . 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 223 . . . . . 6  |-  ( ph  ->  A  e.  ~P X
)
22 xkoopn.c . . . . . 6  |-  ( ph  ->  ( Rt  A )  e.  Comp )
23 oveq2 5908 . . . . . . . 8  |-  ( x  =  A  ->  ( Rt  x )  =  ( Rt  A ) )
2423eleq1d 2382 . . . . . . 7  |-  ( x  =  A  ->  (
( Rt  x )  e.  Comp  <->  ( Rt  A )  e.  Comp ) )
2524elrab 2957 . . . . . 6  |-  ( A  e.  { x  e. 
~P X  |  ( Rt  x )  e.  Comp }  <-> 
( A  e.  ~P X  /\  ( Rt  A )  e.  Comp ) )
2621, 22, 25sylanbrc 645 . . . . 5  |-  ( ph  ->  A  e.  { x  e.  ~P X  |  ( Rt  x )  e.  Comp } )
27 xkoopn.u . . . . 5  |-  ( ph  ->  U  e.  S )
28 eqidd 2317 . . . . 5  |-  ( ph  ->  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  U }  =  { f  e.  ( R  Cn  S
)  |  ( f
" A )  C_  U } )
29 imaeq2 5045 . . . . . . . . 9  |-  ( k  =  A  ->  (
f " k )  =  ( f " A ) )
3029sseq1d 3239 . . . . . . . 8  |-  ( k  =  A  ->  (
( f " k
)  C_  v  <->  ( f " A )  C_  v
) )
3130rabbidv 2814 . . . . . . 7  |-  ( k  =  A  ->  { f  e.  ( R  Cn  S )  |  ( f " k ) 
C_  v }  =  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  v } )
3231eqeq2d 2327 . . . . . 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 3234 . . . . . . . 8  |-  ( v  =  U  ->  (
( f " A
)  C_  v  <->  ( f " A )  C_  U
) )
3433rabbidv 2814 . . . . . . 7  |-  ( v  =  U  ->  { f  e.  ( R  Cn  S )  |  ( f " A ) 
C_  v }  =  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  U } )
3534eqeq2d 2327 . . . . . 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 2926 . . . . 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 1182 . . . 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 4202 . . . . 5  |-  { f  e.  ( R  Cn  S )  |  ( f " A ) 
C_  U }  e.  _V
39 eqeq1 2322 . . . . . 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 2620 . . . . 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 5996 . . . . 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 2951 . . . 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 203 . . 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 3212 . 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 17338 . . 3  |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( S  ^ k o  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 642 . 2  |-  ( ph  ->  ( S  ^ k o  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 2393 1  |-  ( ph  ->  { f  e.  ( R  Cn  S )  |  ( f " A )  C_  U }  e.  ( S  ^ k o  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1633    e. wcel 1701   E.wrex 2578   {crab 2581   _Vcvv 2822    C_ wss 3186   ~Pcpw 3659   U.cuni 3864    X. cxp 4724   ran crn 4727   "cima 4729   -->wf 5288   ` cfv 5292  (class class class)co 5900    e. cmpt2 5902   ficfi 7209   ↾t crest 13374   topGenctg 13391   Topctop 16687    Cn ccn 17010   Compccmp 17169    ^ k o cxko 17312
This theorem is referenced by:  xkouni  17350  xkohaus  17403  xkoptsub  17404  xkoco1cn  17407  xkoco2cn  17408  xkococnlem  17409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-1o 6521  df-en 6907  df-fin 6910  df-fi 7210  df-topgen 13393  df-top 16692  df-xko 17314
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