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Theorem r0cld 17772
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 17759 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
323ad2ant1 979 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  F  Fn  X
)
4 fncnvima2 5854 . . . 4  |-  ( F  Fn  X  ->  ( `' F " { ( F `  A ) } )  =  {
z  e.  X  | 
( F `  z
)  e.  { ( F `  A ) } } )
53, 4syl 16 . . 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 5744 . . . . . 6  |-  ( F `
 z )  e. 
_V
76elsnc 3839 . . . . 5  |-  ( ( F `  z )  e.  { ( F `
 A ) }  <-> 
( F `  z
)  =  ( F `
 A ) )
8 simpl1 961 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  /\  z  e.  X
)  ->  J  e.  (TopOn `  X ) )
9 simpr 449 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  /\  z  e.  X
)  ->  z  e.  X )
10 simpl3 963 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  /\  z  e.  X
)  ->  A  e.  X )
111kqfeq 17758 . . . . . . 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 2499 . . . . . . . . 9  |-  ( y  =  o  ->  (
z  e.  y  <->  z  e.  o ) )
13 eleq2 2499 . . . . . . . . 9  |-  ( y  =  o  ->  ( A  e.  y  <->  A  e.  o ) )
1412, 13bibi12d 314 . . . . . . . 8  |-  ( y  =  o  ->  (
( z  e.  y  <-> 
A  e.  y )  <-> 
( z  e.  o  <-> 
A  e.  o ) ) )
1514cbvralv 2934 . . . . . . 7  |-  ( A. y  e.  J  (
z  e.  y  <->  A  e.  y )  <->  A. o  e.  J  ( z  e.  o  <->  A  e.  o
) )
1611, 15syl6bb 254 . . . . . 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 1185 . . . . 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 250 . . . 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 2949 . . 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 17762 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  F  e.  ( J  Cn  (KQ `  J ) ) )
22213ad2ant1 979 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  F  e.  ( J  Cn  (KQ `  J ) ) )
23 simp2 959 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  (KQ `  J
)  e.  Fre )
24 simp3 960 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  A  e.  X
)
25 fnfvelrn 5869 . . . . . 6  |-  ( ( F  Fn  X  /\  A  e.  X )  ->  ( F `  A
)  e.  ran  F
)
263, 24, 25syl2anc 644 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  ( F `  A )  e.  ran  F )
271kqtopon 17761 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  e.  (TopOn `  ran  F ) )
28273ad2ant1 979 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  (KQ `  J
)  e.  (TopOn `  ran  F ) )
29 toponuni 16994 . . . . . 6  |-  ( (KQ
`  J )  e.  (TopOn `  ran  F )  ->  ran  F  =  U. (KQ `  J ) )
3028, 29syl 16 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  ran  F  =  U. (KQ `  J ) )
3126, 30eleqtrd 2514 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  ( F `  A )  e.  U. (KQ `  J ) )
32 eqid 2438 . . . . 5  |-  U. (KQ `  J )  =  U. (KQ `  J )
3332t1sncld 17392 . . . 4  |-  ( ( (KQ `  J )  e.  Fre  /\  ( F `  A )  e.  U. (KQ `  J
) )  ->  { ( F `  A ) }  e.  ( Clsd `  (KQ `  J ) ) )
3423, 31, 33syl2anc 644 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  { ( F `
 A ) }  e.  ( Clsd `  (KQ `  J ) ) )
35 cnclima 17334 . . 3  |-  ( ( F  e.  ( J  Cn  (KQ `  J
) )  /\  {
( F `  A
) }  e.  (
Clsd `  (KQ `  J
) ) )  -> 
( `' F " { ( F `  A ) } )  e.  ( Clsd `  J
) )
3622, 34, 35syl2anc 644 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  ( `' F " { ( F `  A ) } )  e.  ( Clsd `  J
) )
3720, 36eqeltrrd 2513 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711   {csn 3816   U.cuni 4017    e. cmpt 4268   `'ccnv 4879   ran crn 4881   "cima 4883    Fn wfn 5451   ` cfv 5456  (class class class)co 6083  TopOnctopon 16961   Clsdccld 17082    Cn ccn 17290   Frect1 17373  KQckq 17727
This theorem is referenced by:  nrmr0reg  17783
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-map 7022  df-qtop 13735  df-top 16965  df-topon 16968  df-cld 17085  df-cn 17293  df-t1 17380  df-kq 17728
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