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Theorem r0cld 20002
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 19989 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
323ad2ant1 1017 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  F  Fn  X
)
4 fncnvima2 6003 . . . 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 5876 . . . . . 6  |-  ( F `
 z )  e. 
_V
76elsnc 4051 . . . . 5  |-  ( ( F `  z )  e.  { ( F `
 A ) }  <-> 
( F `  z
)  =  ( F `
 A ) )
8 simpl1 999 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  /\  z  e.  X
)  ->  J  e.  (TopOn `  X ) )
9 simpr 461 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  /\  z  e.  X
)  ->  z  e.  X )
10 simpl3 1001 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  /\  z  e.  X
)  ->  A  e.  X )
111kqfeq 19988 . . . . . . 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 2540 . . . . . . . . 9  |-  ( y  =  o  ->  (
z  e.  y  <->  z  e.  o ) )
13 eleq2 2540 . . . . . . . . 9  |-  ( y  =  o  ->  ( A  e.  y  <->  A  e.  o ) )
1412, 13bibi12d 321 . . . . . . . 8  |-  ( y  =  o  ->  (
( z  e.  y  <-> 
A  e.  y )  <-> 
( z  e.  o  <-> 
A  e.  o ) ) )
1514cbvralv 3088 . . . . . . 7  |-  ( A. y  e.  J  (
z  e.  y  <->  A  e.  y )  <->  A. o  e.  J  ( z  e.  o  <->  A  e.  o
) )
1611, 15syl6bb 261 . . . . . 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 1228 . . . . 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 257 . . . 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 3104 . . 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 2508 . 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 19992 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  F  e.  ( J  Cn  (KQ `  J ) ) )
22213ad2ant1 1017 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  F  e.  ( J  Cn  (KQ `  J ) ) )
23 simp2 997 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  (KQ `  J
)  e.  Fre )
24 simp3 998 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  A  e.  X
)
25 fnfvelrn 6018 . . . . . 6  |-  ( ( F  Fn  X  /\  A  e.  X )  ->  ( F `  A
)  e.  ran  F
)
263, 24, 25syl2anc 661 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  ( F `  A )  e.  ran  F )
271kqtopon 19991 . . . . . . 7  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  e.  (TopOn `  ran  F ) )
28273ad2ant1 1017 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  (KQ `  J
)  e.  (TopOn `  ran  F ) )
29 toponuni 19223 . . . . . 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 2557 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  ( F `  A )  e.  U. (KQ `  J ) )
32 eqid 2467 . . . . 5  |-  U. (KQ `  J )  =  U. (KQ `  J )
3332t1sncld 19621 . . . 4  |-  ( ( (KQ `  J )  e.  Fre  /\  ( F `  A )  e.  U. (KQ `  J
) )  ->  { ( F `  A ) }  e.  ( Clsd `  (KQ `  J ) ) )
3423, 31, 33syl2anc 661 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  { ( F `
 A ) }  e.  ( Clsd `  (KQ `  J ) ) )
35 cnclima 19563 . . 3  |-  ( ( F  e.  ( J  Cn  (KQ `  J
) )  /\  {
( F `  A
) }  e.  (
Clsd `  (KQ `  J
) ) )  -> 
( `' F " { ( F `  A ) } )  e.  ( Clsd `  J
) )
3622, 34, 35syl2anc 661 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  (KQ `  J )  e.  Fre  /\  A  e.  X )  ->  ( `' F " { ( F `  A ) } )  e.  ( Clsd `  J
) )
3720, 36eqeltrrd 2556 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   {crab 2818   {csn 4027   U.cuni 4245    |-> cmpt 4505   `'ccnv 4998   ran crn 5000   "cima 5002    Fn wfn 5583   ` cfv 5588  (class class class)co 6284  TopOnctopon 19190   Clsdccld 19311    Cn ccn 19519   Frect1 19602  KQckq 19957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-map 7422  df-qtop 14762  df-top 19194  df-topon 19197  df-cld 19314  df-cn 19522  df-t1 19609  df-kq 19958
This theorem is referenced by:  nrmr0reg  20013
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