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Theorem kqcldsat 17440
Description: Any closed set is saturated with respect to the topological indistinguishability map (in the terminology of qtoprest 17424). (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 17432 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
3 elpreima 5661 . . . . . 6  |-  ( F  Fn  X  ->  (
z  e.  ( `' F " ( F
" U ) )  <-> 
( z  e.  X  /\  ( F `  z
)  e.  ( F
" U ) ) ) )
42, 3syl 15 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  ( z  e.  ( `' F "
( F " U
) )  <->  ( z  e.  X  /\  ( F `  z )  e.  ( F " U
) ) ) )
54adantr 451 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  (
z  e.  ( `' F " ( F
" U ) )  <-> 
( z  e.  X  /\  ( F `  z
)  e.  ( F
" U ) ) ) )
6 noel 3472 . . . . . . . 8  |-  -.  ( F `  z )  e.  (/)
7 elin 3371 . . . . . . . . 9  |-  ( ( F `  z )  e.  ( ( F
" U )  i^i  ( F " ( X  \  U ) ) )  <->  ( ( F `
 z )  e.  ( F " U
)  /\  ( F `  z )  e.  ( F " ( X 
\  U ) ) ) )
8 incom 3374 . . . . . . . . . . 11  |-  ( ( F " U )  i^i  ( F "
( X  \  U
) ) )  =  ( ( F "
( X  \  U
) )  i^i  ( F " U ) )
9 eqid 2296 . . . . . . . . . . . . . . . . . . . 20  |-  U. J  =  U. J
109cldss 16782 . . . . . . . . . . . . . . . . . . 19  |-  ( U  e.  ( Clsd `  J
)  ->  U  C_  U. J
)
1110adantl 452 . . . . . . . . . . . . . . . . . 18  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  U  C_ 
U. J )
12 fndm 5359 . . . . . . . . . . . . . . . . . . . . 21  |-  ( F  Fn  X  ->  dom  F  =  X )
132, 12syl 15 . . . . . . . . . . . . . . . . . . . 20  |-  ( J  e.  (TopOn `  X
)  ->  dom  F  =  X )
14 toponuni 16681 . . . . . . . . . . . . . . . . . . . 20  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
1513, 14eqtrd 2328 . . . . . . . . . . . . . . . . . . 19  |-  ( J  e.  (TopOn `  X
)  ->  dom  F  = 
U. J )
1615adantr 451 . . . . . . . . . . . . . . . . . 18  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  dom  F  =  U. J )
1711, 16sseqtr4d 3228 . . . . . . . . . . . . . . . . 17  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  U  C_ 
dom  F )
1813adantr 451 . . . . . . . . . . . . . . . . 17  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  dom  F  =  X )
1917, 18sseqtrd 3227 . . . . . . . . . . . . . . . 16  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  U  C_  X )
2019adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  U  C_  X )
21 dfss4 3416 . . . . . . . . . . . . . . 15  |-  ( U 
C_  X  <->  ( X  \  ( X  \  U
) )  =  U )
2220, 21sylib 188 . . . . . . . . . . . . . 14  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  ( X  \  ( X  \  U ) )  =  U )
2322imaeq2d 5028 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  ( F " ( X  \ 
( X  \  U
) ) )  =  ( F " U
) )
2423ineq2d 3383 . . . . . . . . . . . 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 730 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  J  e.  (TopOn `  X )
)
2614adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  X  =  U. J )
2726difeq1d 3306 . . . . . . . . . . . . . . 15  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( X  \  U )  =  ( U. J  \  U ) )
289cldopn 16784 . . . . . . . . . . . . . . . 16  |-  ( U  e.  ( Clsd `  J
)  ->  ( U. J  \  U )  e.  J )
2928adantl 452 . . . . . . . . . . . . . . 15  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( U. J  \  U )  e.  J )
3027, 29eqeltrd 2370 . . . . . . . . . . . . . 14  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( X  \  U )  e.  J )
3130adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  ( X  \  U )  e.  J )
321kqdisj 17439 . . . . . . . . . . . . 13  |-  ( ( J  e.  (TopOn `  X )  /\  ( X  \  U )  e.  J )  ->  (
( F " ( X  \  U ) )  i^i  ( F "
( X  \  ( X  \  U ) ) ) )  =  (/) )
3325, 31, 32syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
( F " ( X  \  U ) )  i^i  ( F "
( X  \  ( X  \  U ) ) ) )  =  (/) )
3424, 33eqtr3d 2330 . . . . . . . . . . 11  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
( F " ( X  \  U ) )  i^i  ( F " U ) )  =  (/) )
358, 34syl5eq 2340 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
( F " U
)  i^i  ( F " ( X  \  U
) ) )  =  (/) )
3635eleq2d 2363 . . . . . . . . 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 250 . . . . . . . 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 294 . . . . . . 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 411 . . . . . . 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 203 . . . . . 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 3175 . . . . . . . . . 10  |-  ( z  e.  ( X  \  U )  <->  ( z  e.  X  /\  -.  z  e.  U ) )
4241baibr 872 . . . . . . . . 9  |-  ( z  e.  X  ->  ( -.  z  e.  U  <->  z  e.  ( X  \  U ) ) )
4342adantl 452 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  ( -.  z  e.  U  <->  z  e.  ( X  \  U ) ) )
44 simpr 447 . . . . . . . . 9  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  z  e.  X )
451kqfvima 17437 . . . . . . . . 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 1182 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
z  e.  ( X 
\  U )  <->  ( F `  z )  e.  ( F " ( X 
\  U ) ) ) )
4743, 46bitrd 244 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  ( -.  z  e.  U  <->  ( F `  z )  e.  ( F "
( X  \  U
) ) ) )
4847con1bid 320 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  ( -.  ( F `  z
)  e.  ( F
" ( X  \  U ) )  <->  z  e.  U ) )
4940, 48sylibd 205 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  /\  z  e.  X )  ->  (
( F `  z
)  e.  ( F
" U )  -> 
z  e.  U ) )
5049expimpd 586 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  (
( z  e.  X  /\  ( F `  z
)  e.  ( F
" U ) )  ->  z  e.  U
) )
515, 50sylbid 206 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  (
z  e.  ( `' F " ( F
" U ) )  ->  z  e.  U
) )
5251ssrdv 3198 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( `' F " ( F
" U ) ) 
C_  U )
53 dminss 5111 . . 3  |-  ( dom 
F  i^i  U )  C_  ( `' F "
( F " U
) )
54 dfss1 3386 . . . . 5  |-  ( U 
C_  dom  F  <->  ( dom  F  i^i  U )  =  U )
5517, 54sylib 188 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( dom  F  i^i  U )  =  U )
5655sseq1d 3218 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  (
( dom  F  i^i  U )  C_  ( `' F " ( F " U ) )  <->  U  C_  ( `' F " ( F
" U ) ) ) )
5753, 56mpbii 202 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  U  C_  ( `' F "
( F " U
) ) )
5852, 57eqssd 3209 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   U.cuni 3843    e. cmpt 4093   `'ccnv 4704   dom cdm 4705   "cima 4708    Fn wfn 5266   ` cfv 5271  TopOnctopon 16648   Clsdccld 16769
This theorem is referenced by:  kqcld  17442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-top 16652  df-topon 16655  df-cld 16772
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