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Theorem r0cld 17445
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 17432 . . . . 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 5663 . . . 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 5555 . . . . . 6  |-  ( F `
 z )  e. 
_V
76elsnc 3676 . . . . 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 17431 . . . . . . 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 2357 . . . . . . . . 9  |-  ( y  =  o  ->  (
z  e.  y  <->  z  e.  o ) )
13 eleq2 2357 . . . . . . . . 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 2777 . . . . . . 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 2792 . . 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 2328 . 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 17435 . . . 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 5678 . . . . . 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 17434 . . . . . . 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 16681 . . . . . 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 2372 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  ( F `  A )  e.  U. (KQ `  J ) )
32 eqid 2296 . . . . 5  |-  U. (KQ `  J )  =  U. (KQ `  J )
3332t1sncld 17070 . . . 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 17013 . . 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 2371 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 1632    e. wcel 1696   A.wral 2556   {crab 2560   {csn 3653   U.cuni 3843    e. cmpt 4093   `'ccnv 4704   ran crn 4706   "cima 4708    Fn wfn 5266   ` cfv 5271  (class class class)co 5874  TopOnctopon 16648   Clsdccld 16769    Cn ccn 16970   Frect1 17051  KQckq 17400
This theorem is referenced by:  nrmr0reg  17456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-qtop 13426  df-top 16652  df-topon 16655  df-cld 16772  df-cn 16973  df-t1 17058  df-kq 17401
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