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Theorem kqopn 17687
Description: The topological indistinguishability map is an open map. (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
kqopn  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( F " U )  e.  (KQ `  J ) )
Distinct variable groups:    x, y, J    x, X, y
Allowed substitution hints:    U( x, y)    F( x, y)

Proof of Theorem kqopn
StepHypRef Expression
1 imassrn 5156 . . . 4  |-  ( F
" U )  C_  ran  F
21a1i 11 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( F " U )  C_  ran  F )
3 kqval.2 . . . . 5  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
43kqsat 17684 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( `' F " ( F
" U ) )  =  U )
5 simpr 448 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  U  e.  J )
64, 5eqeltrd 2461 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( `' F " ( F
" U ) )  e.  J )
73kqffn 17678 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
8 dffn4 5599 . . . . . 6  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
97, 8sylib 189 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  F : X -onto-> ran  F )
109adantr 452 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  F : X -onto-> ran  F )
11 elqtop3 17656 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  F : X -onto-> ran  F )  -> 
( ( F " U )  e.  ( J qTop  F )  <->  ( ( F " U )  C_  ran  F  /\  ( `' F " ( F
" U ) )  e.  J ) ) )
1210, 11syldan 457 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  (
( F " U
)  e.  ( J qTop 
F )  <->  ( ( F " U )  C_  ran  F  /\  ( `' F " ( F
" U ) )  e.  J ) ) )
132, 6, 12mpbir2and 889 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( F " U )  e.  ( J qTop  F ) )
143kqval 17679 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  =  ( J qTop 
F ) )
1514adantr 452 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  (KQ `  J )  =  ( J qTop  F ) )
1613, 15eleqtrrd 2464 1  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( F " U )  e.  (KQ `  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   {crab 2653    C_ wss 3263    e. cmpt 4207   `'ccnv 4817   ran crn 4819   "cima 4821    Fn wfn 5389   -onto->wfo 5392   ` cfv 5394  (class class class)co 6020   qTop cqtop 13656  TopOnctopon 16882  KQckq 17646
This theorem is referenced by:  kqt0lem  17689  isr0  17690  regr1lem  17692  kqreglem1  17694  kqreglem2  17695  kqnrmlem1  17696  kqnrmlem2  17697
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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-qtop 13660  df-topon 16889  df-kq 17647
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