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Theorem r0cld 20688
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 20675 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
323ad2ant1 1026 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  F  Fn  X
)
4 fncnvima2 6019 . . . 4  |-  ( F  Fn  X  ->  ( `' F " { ( F `  A ) } )  =  {
z  e.  X  | 
( F `  z
)  e.  { ( F `  A ) } } )
53, 4syl 17 . . 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 5891 . . . . . 6  |-  ( F `
 z )  e. 
_V
76elsnc 4026 . . . . 5  |-  ( ( F `  z )  e.  { ( F `
 A ) }  <-> 
( F `  z
)  =  ( F `
 A ) )
8 simpl1 1008 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  /\  z  e.  X
)  ->  J  e.  (TopOn `  X ) )
9 simpr 462 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  /\  z  e.  X
)  ->  z  e.  X )
10 simpl3 1010 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  /\  z  e.  X
)  ->  A  e.  X )
111kqfeq 20674 . . . . . . 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 2502 . . . . . . . . 9  |-  ( y  =  o  ->  (
z  e.  y  <->  z  e.  o ) )
13 eleq2 2502 . . . . . . . . 9  |-  ( y  =  o  ->  ( A  e.  y  <->  A  e.  o ) )
1412, 13bibi12d 322 . . . . . . . 8  |-  ( y  =  o  ->  (
( z  e.  y  <-> 
A  e.  y )  <-> 
( z  e.  o  <-> 
A  e.  o ) ) )
1514cbvralv 3062 . . . . . . 7  |-  ( A. y  e.  J  (
z  e.  y  <->  A  e.  y )  <->  A. o  e.  J  ( z  e.  o  <->  A  e.  o
) )
1611, 15syl6bb 264 . . . . . 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 1264 . . . . 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 260 . . . 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 3078 . . 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 2470 . 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 20678 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  F  e.  ( J  Cn  (KQ `  J ) ) )
22213ad2ant1 1026 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  F  e.  ( J  Cn  (KQ `  J ) ) )
23 simp2 1006 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  (KQ `  J
)  e.  Fre )
24 simp3 1007 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  A  e.  X
)
25 fnfvelrn 6034 . . . . . 6  |-  ( ( F  Fn  X  /\  A  e.  X )  ->  ( F `  A
)  e.  ran  F
)
263, 24, 25syl2anc 665 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  ( F `  A )  e.  ran  F )
271kqtopon 20677 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  e.  (TopOn `  ran  F ) )
28273ad2ant1 1026 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  (KQ `  J
)  e.  (TopOn `  ran  F ) )
29 toponuni 19877 . . . . . 6  |-  ( (KQ
`  J )  e.  (TopOn `  ran  F )  ->  ran  F  =  U. (KQ `  J ) )
3028, 29syl 17 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  ran  F  =  U. (KQ `  J ) )
3126, 30eleqtrd 2519 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  ( F `  A )  e.  U. (KQ `  J ) )
32 eqid 2429 . . . . 5  |-  U. (KQ `  J )  =  U. (KQ `  J )
3332t1sncld 20277 . . . 4  |-  ( ( (KQ `  J )  e.  Fre  /\  ( F `  A )  e.  U. (KQ `  J
) )  ->  { ( F `  A ) }  e.  ( Clsd `  (KQ `  J ) ) )
3423, 31, 33syl2anc 665 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  { ( F `
 A ) }  e.  ( Clsd `  (KQ `  J ) ) )
35 cnclima 20219 . . 3  |-  ( ( F  e.  ( J  Cn  (KQ `  J
) )  /\  {
( F `  A
) }  e.  (
Clsd `  (KQ `  J
) ) )  -> 
( `' F " { ( F `  A ) } )  e.  ( Clsd `  J
) )
3622, 34, 35syl2anc 665 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  ( `' F " { ( F `  A ) } )  e.  ( Clsd `  J
) )
3720, 36eqeltrrd 2518 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 setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782   {crab 2786   {csn 4002   U.cuni 4222    |-> cmpt 4484   `'ccnv 4853   ran crn 4855   "cima 4857    Fn wfn 5596   ` cfv 5601  (class class class)co 6305  TopOnctopon 19853   Clsdccld 19966    Cn ccn 20175   Frect1 20258  KQckq 20643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-map 7482  df-qtop 15368  df-top 19856  df-topon 19858  df-cld 19969  df-cn 20178  df-t1 20265  df-kq 20644
This theorem is referenced by:  nrmr0reg  20699
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