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Theorem kqopn 17441
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 5041 . . . 4  |-  ( F
" U )  C_  ran  F
21a1i 10 . . 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 17438 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( `' F " ( F
" U ) )  =  U )
5 simpr 447 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  U  e.  J )
64, 5eqeltrd 2370 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( `' F " ( F
" U ) )  e.  J )
73kqffn 17432 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
8 dffn4 5473 . . . . . 6  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
97, 8sylib 188 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  F : X -onto-> ran  F )
109adantr 451 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  F : X -onto-> ran  F )
11 elqtop3 17410 . . . 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 456 . . 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 888 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( F " U )  e.  ( J qTop  F ) )
143kqval 17433 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  =  ( J qTop 
F ) )
1514adantr 451 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  (KQ `  J )  =  ( J qTop  F ) )
1613, 15eleqtrrd 2373 1  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( F " U )  e.  (KQ `  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {crab 2560    C_ wss 3165    e. cmpt 4093   `'ccnv 4704   ran crn 4706   "cima 4708    Fn wfn 5266   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874   qTop cqtop 13422  TopOnctopon 16648  KQckq 17400
This theorem is referenced by:  kqt0lem  17443  isr0  17444  regr1lem  17446  kqreglem1  17448  kqreglem2  17449  kqnrmlem1  17450  kqnrmlem2  17451
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-rep 4147  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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-iun 3923  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-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-qtop 13426  df-topon 16655  df-kq 17401
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