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Theorem kqcldsat 17767
Description: Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 17751). (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 17759 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
3 elpreima 5852 . . . . . 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 453 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  (
z  e.  ( `' F " ( F
" U ) )  <-> 
( z  e.  X  /\  ( F `  z
)  e.  ( F
" U ) ) ) )
6 noel 3634 . . . . . . . 8  |-  -.  ( F `  z )  e.  (/)
7 elin 3532 . . . . . . . . 9  |-  ( ( F `  z )  e.  ( ( F
" U )  i^i  ( F " ( X  \  U ) ) )  <->  ( ( F `
 z )  e.  ( F " U
)  /\  ( F `  z )  e.  ( F " ( X 
\  U ) ) ) )
8 incom 3535 . . . . . . . . . . 11  |-  ( ( F " U )  i^i  ( F "
( X  \  U
) ) )  =  ( ( F "
( X  \  U
) )  i^i  ( F " U ) )
9 eqid 2438 . . . . . . . . . . . . . . . . . . . 20  |-  U. J  =  U. J
109cldss 17095 . . . . . . . . . . . . . . . . . . 19  |-  ( U  e.  ( Clsd `  J
)  ->  U  C_  U. J
)
1110adantl 454 . . . . . . . . . . . . . . . . . 18  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  U  C_ 
U. J )
12 fndm 5546 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F  Fn  X  ->  dom  F  =  X )
132, 12syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( J  e.  (TopOn `  X
)  ->  dom  F  =  X )
14 toponuni 16994 . . . . . . . . . . . . . . . . . . . 20  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
1513, 14eqtrd 2470 . . . . . . . . . . . . . . . . . . 19  |-  ( J  e.  (TopOn `  X
)  ->  dom  F  = 
U. J )
1615adantr 453 . . . . . . . . . . . . . . . . . 18  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  dom  F  =  U. J )
1711, 16sseqtr4d 3387 . . . . . . . . . . . . . . . . 17  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  U  C_ 
dom  F )
1813adantr 453 . . . . . . . . . . . . . . . . 17  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  dom  F  =  X )
1917, 18sseqtrd 3386 . . . . . . . . . . . . . . . 16  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  U  C_  X )
2019adantr 453 . . . . . . . . . . . . . . 15  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  U  C_  X )
21 dfss4 3577 . . . . . . . . . . . . . . 15  |-  ( U 
C_  X  <->  ( X  \  ( X  \  U
) )  =  U )
2220, 21sylib 190 . . . . . . . . . . . . . 14  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  ( X  \  ( X  \  U ) )  =  U )
2322imaeq2d 5205 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  ( F " ( X  \ 
( X  \  U
) ) )  =  ( F " U
) )
2423ineq2d 3544 . . . . . . . . . . . 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 732 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  J  e.  (TopOn `  X )
)
2614adantr 453 . . . . . . . . . . . . . . . 16  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  X  =  U. J )
2726difeq1d 3466 . . . . . . . . . . . . . . 15  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( X  \  U )  =  ( U. J  \  U ) )
289cldopn 17097 . . . . . . . . . . . . . . . 16  |-  ( U  e.  ( Clsd `  J
)  ->  ( U. J  \  U )  e.  J )
2928adantl 454 . . . . . . . . . . . . . . 15  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( U. J  \  U )  e.  J )
3027, 29eqeltrd 2512 . . . . . . . . . . . . . 14  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( X  \  U )  e.  J )
3130adantr 453 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  ( X  \  U )  e.  J )
321kqdisj 17766 . . . . . . . . . . . . 13  |-  ( ( J  e.  (TopOn `  X )  /\  ( X  \  U )  e.  J )  ->  (
( F " ( X  \  U ) )  i^i  ( F "
( X  \  ( X  \  U ) ) ) )  =  (/) )
3325, 31, 32syl2anc 644 . . . . . . . . . . . 12  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
( F " ( X  \  U ) )  i^i  ( F "
( X  \  ( X  \  U ) ) ) )  =  (/) )
3424, 33eqtr3d 2472 . . . . . . . . . . 11  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
( F " ( X  \  U ) )  i^i  ( F " U ) )  =  (/) )
358, 34syl5eq 2482 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
( F " U
)  i^i  ( F " ( X  \  U
) ) )  =  (/) )
3635eleq2d 2505 . . . . . . . . 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 252 . . . . . . . 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 296 . . . . . . 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 413 . . . . . . 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 205 . . . . . 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 3332 . . . . . . . . . 10  |-  ( z  e.  ( X  \  U )  <->  ( z  e.  X  /\  -.  z  e.  U ) )
4241baibr 874 . . . . . . . . 9  |-  ( z  e.  X  ->  ( -.  z  e.  U  <->  z  e.  ( X  \  U ) ) )
4342adantl 454 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  ( -.  z  e.  U  <->  z  e.  ( X  \  U ) ) )
44 simpr 449 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  z  e.  X )
451kqfvima 17764 . . . . . . . . 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 1185 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
z  e.  ( X 
\  U )  <->  ( F `  z )  e.  ( F " ( X 
\  U ) ) ) )
4743, 46bitrd 246 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  ( -.  z  e.  U  <->  ( F `  z )  e.  ( F "
( X  \  U
) ) ) )
4847con1bid 322 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  ( -.  ( F `  z
)  e.  ( F
" ( X  \  U ) )  <->  z  e.  U ) )
4940, 48sylibd 207 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
( F `  z
)  e.  ( F
" U )  -> 
z  e.  U ) )
5049expimpd 588 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  (
( z  e.  X  /\  ( F `  z
)  e.  ( F
" U ) )  ->  z  e.  U
) )
515, 50sylbid 208 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  (
z  e.  ( `' F " ( F
" U ) )  ->  z  e.  U
) )
5251ssrdv 3356 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( `' F " ( F
" U ) ) 
C_  U )
53 dfss1 3547 . . . 4  |-  ( U 
C_  dom  F  <->  ( dom  F  i^i  U )  =  U )
5417, 53sylib 190 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( dom  F  i^i  U )  =  U )
55 dminss 5288 . . 3  |-  ( dom 
F  i^i  U )  C_  ( `' F "
( F " U
) )
5654, 55syl6eqssr 3401 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  U  C_  ( `' F "
( F " U
) ) )
5752, 56eqssd 3367 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2711    \ cdif 3319    i^i cin 3321    C_ wss 3322   (/)c0 3630   U.cuni 4017    e. cmpt 4268   `'ccnv 4879   dom cdm 4880   "cima 4883    Fn wfn 5451   ` cfv 5456  TopOnctopon 16961   Clsdccld 17082
This theorem is referenced by:  kqcld  17769
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-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-rab 2716  df-v 2960  df-sbc 3164  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-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-fv 5464  df-top 16965  df-topon 16968  df-cld 17085
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