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Theorem kqcldsat 20524
Description: Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 20508). (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 20516 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
3 elpreima 5984 . . . . . 6  |-  ( F  Fn  X  ->  (
z  e.  ( `' F " ( F
" U ) )  <-> 
( z  e.  X  /\  ( F `  z
)  e.  ( F
" U ) ) ) )
42, 3syl 17 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  ( z  e.  ( `' F "
( F " U
) )  <->  ( z  e.  X  /\  ( F `  z )  e.  ( F " U
) ) ) )
54adantr 463 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  (
z  e.  ( `' F " ( F
" U ) )  <-> 
( z  e.  X  /\  ( F `  z
)  e.  ( F
" U ) ) ) )
6 noel 3741 . . . . . . . 8  |-  -.  ( F `  z )  e.  (/)
7 elin 3625 . . . . . . . . 9  |-  ( ( F `  z )  e.  ( ( F
" U )  i^i  ( F " ( X  \  U ) ) )  <->  ( ( F `
 z )  e.  ( F " U
)  /\  ( F `  z )  e.  ( F " ( X 
\  U ) ) ) )
8 incom 3631 . . . . . . . . . . 11  |-  ( ( F " U )  i^i  ( F "
( X  \  U
) ) )  =  ( ( F "
( X  \  U
) )  i^i  ( F " U ) )
9 eqid 2402 . . . . . . . . . . . . . . . . . . . 20  |-  U. J  =  U. J
109cldss 19820 . . . . . . . . . . . . . . . . . . 19  |-  ( U  e.  ( Clsd `  J
)  ->  U  C_  U. J
)
1110adantl 464 . . . . . . . . . . . . . . . . . 18  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  U  C_ 
U. J )
12 fndm 5660 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F  Fn  X  ->  dom  F  =  X )
132, 12syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( J  e.  (TopOn `  X
)  ->  dom  F  =  X )
14 toponuni 19718 . . . . . . . . . . . . . . . . . . . 20  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
1513, 14eqtrd 2443 . . . . . . . . . . . . . . . . . . 19  |-  ( J  e.  (TopOn `  X
)  ->  dom  F  = 
U. J )
1615adantr 463 . . . . . . . . . . . . . . . . . 18  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  dom  F  =  U. J )
1711, 16sseqtr4d 3478 . . . . . . . . . . . . . . . . 17  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  U  C_ 
dom  F )
1813adantr 463 . . . . . . . . . . . . . . . . 17  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  dom  F  =  X )
1917, 18sseqtrd 3477 . . . . . . . . . . . . . . . 16  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  U  C_  X )
2019adantr 463 . . . . . . . . . . . . . . 15  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  U  C_  X )
21 dfss4 3683 . . . . . . . . . . . . . . 15  |-  ( U 
C_  X  <->  ( X  \  ( X  \  U
) )  =  U )
2220, 21sylib 196 . . . . . . . . . . . . . 14  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  ( X  \  ( X  \  U ) )  =  U )
2322imaeq2d 5156 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  ( F " ( X  \ 
( X  \  U
) ) )  =  ( F " U
) )
2423ineq2d 3640 . . . . . . . . . . . 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 752 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  J  e.  (TopOn `  X )
)
2614adantr 463 . . . . . . . . . . . . . . . 16  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  X  =  U. J )
2726difeq1d 3559 . . . . . . . . . . . . . . 15  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( X  \  U )  =  ( U. J  \  U ) )
289cldopn 19822 . . . . . . . . . . . . . . . 16  |-  ( U  e.  ( Clsd `  J
)  ->  ( U. J  \  U )  e.  J )
2928adantl 464 . . . . . . . . . . . . . . 15  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( U. J  \  U )  e.  J )
3027, 29eqeltrd 2490 . . . . . . . . . . . . . 14  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( X  \  U )  e.  J )
3130adantr 463 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  ( X  \  U )  e.  J )
321kqdisj 20523 . . . . . . . . . . . . 13  |-  ( ( J  e.  (TopOn `  X )  /\  ( X  \  U )  e.  J )  ->  (
( F " ( X  \  U ) )  i^i  ( F "
( X  \  ( X  \  U ) ) ) )  =  (/) )
3325, 31, 32syl2anc 659 . . . . . . . . . . . 12  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
( F " ( X  \  U ) )  i^i  ( F "
( X  \  ( X  \  U ) ) ) )  =  (/) )
3424, 33eqtr3d 2445 . . . . . . . . . . 11  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
( F " ( X  \  U ) )  i^i  ( F " U ) )  =  (/) )
358, 34syl5eq 2455 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
( F " U
)  i^i  ( F " ( X  \  U
) ) )  =  (/) )
3635eleq2d 2472 . . . . . . . . 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 259 . . . . . . . 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 301 . . . . . . 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 420 . . . . . . 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 212 . . . . . 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 3423 . . . . . . . . . 10  |-  ( z  e.  ( X  \  U )  <->  ( z  e.  X  /\  -.  z  e.  U ) )
4241baibr 905 . . . . . . . . 9  |-  ( z  e.  X  ->  ( -.  z  e.  U  <->  z  e.  ( X  \  U ) ) )
4342adantl 464 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  ( -.  z  e.  U  <->  z  e.  ( X  \  U ) ) )
44 simpr 459 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  z  e.  X )
451kqfvima 20521 . . . . . . . . 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 1230 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
z  e.  ( X 
\  U )  <->  ( F `  z )  e.  ( F " ( X 
\  U ) ) ) )
4743, 46bitrd 253 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  ( -.  z  e.  U  <->  ( F `  z )  e.  ( F "
( X  \  U
) ) ) )
4847con1bid 328 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  ( -.  ( F `  z
)  e.  ( F
" ( X  \  U ) )  <->  z  e.  U ) )
4940, 48sylibd 214 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
( F `  z
)  e.  ( F
" U )  -> 
z  e.  U ) )
5049expimpd 601 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  (
( z  e.  X  /\  ( F `  z
)  e.  ( F
" U ) )  ->  z  e.  U
) )
515, 50sylbid 215 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  (
z  e.  ( `' F " ( F
" U ) )  ->  z  e.  U
) )
5251ssrdv 3447 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( `' F " ( F
" U ) ) 
C_  U )
53 dfss1 3643 . . . 4  |-  ( U 
C_  dom  F  <->  ( dom  F  i^i  U )  =  U )
5417, 53sylib 196 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( dom  F  i^i  U )  =  U )
55 dminss 5237 . . 3  |-  ( dom 
F  i^i  U )  C_  ( `' F "
( F " U
) )
5654, 55syl6eqssr 3492 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  U  C_  ( `' F "
( F " U
) ) )
5752, 56eqssd 3458 1  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( `' F " ( F
" U ) )  =  U )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   {crab 2757    \ cdif 3410    i^i cin 3412    C_ wss 3413   (/)c0 3737   U.cuni 4190    |-> cmpt 4452   `'ccnv 4821   dom cdm 4822   "cima 4825    Fn wfn 5563   ` cfv 5568  TopOnctopon 19685   Clsdccld 19807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fv 5576  df-top 19689  df-topon 19692  df-cld 19810
This theorem is referenced by:  kqcld  20526
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