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Theorem kqcldsat 17688
Description: Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 17672). (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
kqcldsat  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( `' F " ( F
" U ) )  =  U )
Distinct variable groups:    x, y, J    x, X, y
Allowed substitution hints:    U( x, y)    F( x, y)

Proof of Theorem kqcldsat
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 kqval.2 . . . . . . 7  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
21kqffn 17680 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
3 elpreima 5791 . . . . . 6  |-  ( F  Fn  X  ->  (
z  e.  ( `' F " ( F
" U ) )  <-> 
( z  e.  X  /\  ( F `  z
)  e.  ( F
" U ) ) ) )
42, 3syl 16 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  ( z  e.  ( `' F "
( F " U
) )  <->  ( z  e.  X  /\  ( F `  z )  e.  ( F " U
) ) ) )
54adantr 452 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  (
z  e.  ( `' F " ( F
" U ) )  <-> 
( z  e.  X  /\  ( F `  z
)  e.  ( F
" U ) ) ) )
6 noel 3577 . . . . . . . 8  |-  -.  ( F `  z )  e.  (/)
7 elin 3475 . . . . . . . . 9  |-  ( ( F `  z )  e.  ( ( F
" U )  i^i  ( F " ( X  \  U ) ) )  <->  ( ( F `
 z )  e.  ( F " U
)  /\  ( F `  z )  e.  ( F " ( X 
\  U ) ) ) )
8 incom 3478 . . . . . . . . . . 11  |-  ( ( F " U )  i^i  ( F "
( X  \  U
) ) )  =  ( ( F "
( X  \  U
) )  i^i  ( F " U ) )
9 eqid 2389 . . . . . . . . . . . . . . . . . . . 20  |-  U. J  =  U. J
109cldss 17018 . . . . . . . . . . . . . . . . . . 19  |-  ( U  e.  ( Clsd `  J
)  ->  U  C_  U. J
)
1110adantl 453 . . . . . . . . . . . . . . . . . 18  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  U  C_ 
U. J )
12 fndm 5486 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F  Fn  X  ->  dom  F  =  X )
132, 12syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( J  e.  (TopOn `  X
)  ->  dom  F  =  X )
14 toponuni 16917 . . . . . . . . . . . . . . . . . . . 20  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
1513, 14eqtrd 2421 . . . . . . . . . . . . . . . . . . 19  |-  ( J  e.  (TopOn `  X
)  ->  dom  F  = 
U. J )
1615adantr 452 . . . . . . . . . . . . . . . . . 18  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  dom  F  =  U. J )
1711, 16sseqtr4d 3330 . . . . . . . . . . . . . . . . 17  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  U  C_ 
dom  F )
1813adantr 452 . . . . . . . . . . . . . . . . 17  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  dom  F  =  X )
1917, 18sseqtrd 3329 . . . . . . . . . . . . . . . 16  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  U  C_  X )
2019adantr 452 . . . . . . . . . . . . . . 15  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  U  C_  X )
21 dfss4 3520 . . . . . . . . . . . . . . 15  |-  ( U 
C_  X  <->  ( X  \  ( X  \  U
) )  =  U )
2220, 21sylib 189 . . . . . . . . . . . . . 14  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  ( X  \  ( X  \  U ) )  =  U )
2322imaeq2d 5145 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  ( F " ( X  \ 
( X  \  U
) ) )  =  ( F " U
) )
2423ineq2d 3487 . . . . . . . . . . . 12  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
( F " ( X  \  U ) )  i^i  ( F "
( X  \  ( X  \  U ) ) ) )  =  ( ( F " ( X  \  U ) )  i^i  ( F " U ) ) )
25 simpll 731 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  J  e.  (TopOn `  X )
)
2614adantr 452 . . . . . . . . . . . . . . . 16  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  X  =  U. J )
2726difeq1d 3409 . . . . . . . . . . . . . . 15  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( X  \  U )  =  ( U. J  \  U ) )
289cldopn 17020 . . . . . . . . . . . . . . . 16  |-  ( U  e.  ( Clsd `  J
)  ->  ( U. J  \  U )  e.  J )
2928adantl 453 . . . . . . . . . . . . . . 15  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( U. J  \  U )  e.  J )
3027, 29eqeltrd 2463 . . . . . . . . . . . . . 14  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( X  \  U )  e.  J )
3130adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  ( X  \  U )  e.  J )
321kqdisj 17687 . . . . . . . . . . . . 13  |-  ( ( J  e.  (TopOn `  X )  /\  ( X  \  U )  e.  J )  ->  (
( F " ( X  \  U ) )  i^i  ( F "
( X  \  ( X  \  U ) ) ) )  =  (/) )
3325, 31, 32syl2anc 643 . . . . . . . . . . . 12  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
( F " ( X  \  U ) )  i^i  ( F "
( X  \  ( X  \  U ) ) ) )  =  (/) )
3424, 33eqtr3d 2423 . . . . . . . . . . 11  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
( F " ( X  \  U ) )  i^i  ( F " U ) )  =  (/) )
358, 34syl5eq 2433 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
( F " U
)  i^i  ( F " ( X  \  U
) ) )  =  (/) )
3635eleq2d 2456 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
( F `  z
)  e.  ( ( F " U )  i^i  ( F "
( X  \  U
) ) )  <->  ( F `  z )  e.  (/) ) )
377, 36syl5bbr 251 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
( ( F `  z )  e.  ( F " U )  /\  ( F `  z )  e.  ( F " ( X 
\  U ) ) )  <->  ( F `  z )  e.  (/) ) )
386, 37mtbiri 295 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  -.  ( ( F `  z )  e.  ( F " U )  /\  ( F `  z )  e.  ( F " ( X 
\  U ) ) ) )
39 imnan 412 . . . . . . 7  |-  ( ( ( F `  z
)  e.  ( F
" U )  ->  -.  ( F `  z
)  e.  ( F
" ( X  \  U ) ) )  <->  -.  ( ( F `  z )  e.  ( F " U )  /\  ( F `  z )  e.  ( F " ( X 
\  U ) ) ) )
4038, 39sylibr 204 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
( F `  z
)  e.  ( F
" U )  ->  -.  ( F `  z
)  e.  ( F
" ( X  \  U ) ) ) )
41 eldif 3275 . . . . . . . . . 10  |-  ( z  e.  ( X  \  U )  <->  ( z  e.  X  /\  -.  z  e.  U ) )
4241baibr 873 . . . . . . . . 9  |-  ( z  e.  X  ->  ( -.  z  e.  U  <->  z  e.  ( X  \  U ) ) )
4342adantl 453 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  ( -.  z  e.  U  <->  z  e.  ( X  \  U ) ) )
44 simpr 448 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  z  e.  X )
451kqfvima 17685 . . . . . . . . 9  |-  ( ( J  e.  (TopOn `  X )  /\  ( X  \  U )  e.  J  /\  z  e.  X )  ->  (
z  e.  ( X 
\  U )  <->  ( F `  z )  e.  ( F " ( X 
\  U ) ) ) )
4625, 31, 44, 45syl3anc 1184 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
z  e.  ( X 
\  U )  <->  ( F `  z )  e.  ( F " ( X 
\  U ) ) ) )
4743, 46bitrd 245 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  ( -.  z  e.  U  <->  ( F `  z )  e.  ( F "
( X  \  U
) ) ) )
4847con1bid 321 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  ( -.  ( F `  z
)  e.  ( F
" ( X  \  U ) )  <->  z  e.  U ) )
4940, 48sylibd 206 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
( F `  z
)  e.  ( F
" U )  -> 
z  e.  U ) )
5049expimpd 587 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  (
( z  e.  X  /\  ( F `  z
)  e.  ( F
" U ) )  ->  z  e.  U
) )
515, 50sylbid 207 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  (
z  e.  ( `' F " ( F
" U ) )  ->  z  e.  U
) )
5251ssrdv 3299 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( `' F " ( F
" U ) ) 
C_  U )
53 dfss1 3490 . . . 4  |-  ( U 
C_  dom  F  <->  ( dom  F  i^i  U )  =  U )
5417, 53sylib 189 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( dom  F  i^i  U )  =  U )
55 dminss 5228 . . 3  |-  ( dom 
F  i^i  U )  C_  ( `' F "
( F " U
) )
5654, 55syl6eqssr 3344 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  U  C_  ( `' F "
( F " U
) ) )
5752, 56eqssd 3310 1  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( `' F " ( F
" U ) )  =  U )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   {crab 2655    \ cdif 3262    i^i cin 3264    C_ wss 3265   (/)c0 3573   U.cuni 3959    e. cmpt 4209   `'ccnv 4819   dom cdm 4820   "cima 4823    Fn wfn 5391   ` cfv 5396  TopOnctopon 16884   Clsdccld 17005
This theorem is referenced by:  kqcld  17690
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fv 5404  df-top 16888  df-topon 16891  df-cld 17008
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