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Theorem r0cld 17429
Description: The analogue of the T1 axiom (singletons are closed) for an R0 space. In an R0 space the set of all points topologically indistinguishable from  A is closed. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
Assertion
Ref Expression
r0cld  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  { z  e.  X  |  A. o  e.  J  ( z  e.  o  <->  A  e.  o
) }  e.  (
Clsd `  J )
)
Distinct variable groups:    x, o,
y, z, A    o, J, x, y, z    o, F, z    o, X, x, y, z
Allowed substitution hints:    F( x, y)

Proof of Theorem r0cld
StepHypRef Expression
1 kqval.2 . . . . . 6  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
21kqffn 17416 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
323ad2ant1 976 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  F  Fn  X
)
4 fncnvima2 5647 . . . 4  |-  ( F  Fn  X  ->  ( `' F " { ( F `  A ) } )  =  {
z  e.  X  | 
( F `  z
)  e.  { ( F `  A ) } } )
53, 4syl 15 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  ( `' F " { ( F `  A ) } )  =  { z  e.  X  |  ( F `
 z )  e. 
{ ( F `  A ) } }
)
6 fvex 5539 . . . . . 6  |-  ( F `
 z )  e. 
_V
76elsnc 3663 . . . . 5  |-  ( ( F `  z )  e.  { ( F `
 A ) }  <-> 
( F `  z
)  =  ( F `
 A ) )
8 simpl1 958 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  /\  z  e.  X
)  ->  J  e.  (TopOn `  X ) )
9 simpr 447 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  /\  z  e.  X
)  ->  z  e.  X )
10 simpl3 960 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  /\  z  e.  X
)  ->  A  e.  X )
111kqfeq 17415 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  z  e.  X  /\  A  e.  X )  ->  (
( F `  z
)  =  ( F `
 A )  <->  A. y  e.  J  ( z  e.  y  <->  A  e.  y
) ) )
12 eleq2 2344 . . . . . . . . 9  |-  ( y  =  o  ->  (
z  e.  y  <->  z  e.  o ) )
13 eleq2 2344 . . . . . . . . 9  |-  ( y  =  o  ->  ( A  e.  y  <->  A  e.  o ) )
1412, 13bibi12d 312 . . . . . . . 8  |-  ( y  =  o  ->  (
( z  e.  y  <-> 
A  e.  y )  <-> 
( z  e.  o  <-> 
A  e.  o ) ) )
1514cbvralv 2764 . . . . . . 7  |-  ( A. y  e.  J  (
z  e.  y  <->  A  e.  y )  <->  A. o  e.  J  ( z  e.  o  <->  A  e.  o
) )
1611, 15syl6bb 252 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  z  e.  X  /\  A  e.  X )  ->  (
( F `  z
)  =  ( F `
 A )  <->  A. o  e.  J  ( z  e.  o  <->  A  e.  o
) ) )
178, 9, 10, 16syl3anc 1182 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  /\  z  e.  X
)  ->  ( ( F `  z )  =  ( F `  A )  <->  A. o  e.  J  ( z  e.  o  <->  A  e.  o
) ) )
187, 17syl5bb 248 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  /\  z  e.  X
)  ->  ( ( F `  z )  e.  { ( F `  A ) }  <->  A. o  e.  J  ( z  e.  o  <->  A  e.  o
) ) )
1918rabbidva 2779 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  { z  e.  X  |  ( F `
 z )  e. 
{ ( F `  A ) } }  =  { z  e.  X  |  A. o  e.  J  ( z  e.  o  <-> 
A  e.  o ) } )
205, 19eqtrd 2315 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  ( `' F " { ( F `  A ) } )  =  { z  e.  X  |  A. o  e.  J  ( z  e.  o  <->  A  e.  o
) } )
211kqid 17419 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  F  e.  ( J  Cn  (KQ `  J ) ) )
22213ad2ant1 976 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  F  e.  ( J  Cn  (KQ `  J ) ) )
23 simp2 956 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  (KQ `  J
)  e.  Fre )
24 simp3 957 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  A  e.  X
)
25 fnfvelrn 5662 . . . . . 6  |-  ( ( F  Fn  X  /\  A  e.  X )  ->  ( F `  A
)  e.  ran  F
)
263, 24, 25syl2anc 642 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  ( F `  A )  e.  ran  F )
271kqtopon 17418 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  e.  (TopOn `  ran  F ) )
28273ad2ant1 976 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  (KQ `  J
)  e.  (TopOn `  ran  F ) )
29 toponuni 16665 . . . . . 6  |-  ( (KQ
`  J )  e.  (TopOn `  ran  F )  ->  ran  F  =  U. (KQ `  J ) )
3028, 29syl 15 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  ran  F  =  U. (KQ `  J ) )
3126, 30eleqtrd 2359 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  ( F `  A )  e.  U. (KQ `  J ) )
32 eqid 2283 . . . . 5  |-  U. (KQ `  J )  =  U. (KQ `  J )
3332t1sncld 17054 . . . 4  |-  ( ( (KQ `  J )  e.  Fre  /\  ( F `  A )  e.  U. (KQ `  J
) )  ->  { ( F `  A ) }  e.  ( Clsd `  (KQ `  J ) ) )
3423, 31, 33syl2anc 642 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  { ( F `
 A ) }  e.  ( Clsd `  (KQ `  J ) ) )
35 cnclima 16997 . . 3  |-  ( ( F  e.  ( J  Cn  (KQ `  J
) )  /\  {
( F `  A
) }  e.  (
Clsd `  (KQ `  J
) ) )  -> 
( `' F " { ( F `  A ) } )  e.  ( Clsd `  J
) )
3622, 34, 35syl2anc 642 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  ( `' F " { ( F `  A ) } )  e.  ( Clsd `  J
) )
3720, 36eqeltrrd 2358 1  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  { z  e.  X  |  A. o  e.  J  ( z  e.  o  <->  A  e.  o
) }  e.  (
Clsd `  J )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   {csn 3640   U.cuni 3827    e. cmpt 4077   `'ccnv 4688   ran crn 4690   "cima 4692    Fn wfn 5250   ` cfv 5255  (class class class)co 5858  TopOnctopon 16632   Clsdccld 16753    Cn ccn 16954   Frect1 17035  KQckq 17384
This theorem is referenced by:  nrmr0reg  17440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-qtop 13410  df-top 16636  df-topon 16639  df-cld 16756  df-cn 16957  df-t1 17042  df-kq 17385
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