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Theorem kqcldsat 17718
Description: Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 17702). (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 17710 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
3 elpreima 5809 . . . . . 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 3592 . . . . . . . 8  |-  -.  ( F `  z )  e.  (/)
7 elin 3490 . . . . . . . . 9  |-  ( ( F `  z )  e.  ( ( F
" U )  i^i  ( F " ( X  \  U ) ) )  <->  ( ( F `
 z )  e.  ( F " U
)  /\  ( F `  z )  e.  ( F " ( X 
\  U ) ) ) )
8 incom 3493 . . . . . . . . . . 11  |-  ( ( F " U )  i^i  ( F "
( X  \  U
) ) )  =  ( ( F "
( X  \  U
) )  i^i  ( F " U ) )
9 eqid 2404 . . . . . . . . . . . . . . . . . . . 20  |-  U. J  =  U. J
109cldss 17048 . . . . . . . . . . . . . . . . . . 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 5503 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F  Fn  X  ->  dom  F  =  X )
132, 12syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( J  e.  (TopOn `  X
)  ->  dom  F  =  X )
14 toponuni 16947 . . . . . . . . . . . . . . . . . . . 20  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
1513, 14eqtrd 2436 . . . . . . . . . . . . . . . . . . 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 3345 . . . . . . . . . . . . . . . . 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 3344 . . . . . . . . . . . . . . . 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 3535 . . . . . . . . . . . . . . 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 5162 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  ( F " ( X  \ 
( X  \  U
) ) )  =  ( F " U
) )
2423ineq2d 3502 . . . . . . . . . . . 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 3424 . . . . . . . . . . . . . . 15  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( X  \  U )  =  ( U. J  \  U ) )
289cldopn 17050 . . . . . . . . . . . . . . . 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 2478 . . . . . . . . . . . . . 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 17717 . . . . . . . . . . . . 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 2438 . . . . . . . . . . 11  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
( F " ( X  \  U ) )  i^i  ( F " U ) )  =  (/) )
358, 34syl5eq 2448 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
( F " U
)  i^i  ( F " ( X  \  U
) ) )  =  (/) )
3635eleq2d 2471 . . . . . . . . 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 3290 . . . . . . . . . 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 17715 . . . . . . . . 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 3314 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( `' F " ( F
" U ) ) 
C_  U )
53 dfss1 3505 . . . 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 5245 . . 3  |-  ( dom 
F  i^i  U )  C_  ( `' F "
( F " U
) )
5654, 55syl6eqssr 3359 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  U  C_  ( `' F "
( F " U
) ) )
5752, 56eqssd 3325 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 1721   {crab 2670    \ cdif 3277    i^i cin 3279    C_ wss 3280   (/)c0 3588   U.cuni 3975    e. cmpt 4226   `'ccnv 4836   dom cdm 4837   "cima 4840    Fn wfn 5408   ` cfv 5413  TopOnctopon 16914   Clsdccld 17035
This theorem is referenced by:  kqcld  17720
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-top 16918  df-topon 16921  df-cld 17038
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