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Theorem kqopn 20103
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 5354 . . . 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 20100 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( `' F " ( F
" U ) )  =  U )
5 simpr 461 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  U  e.  J )
64, 5eqeltrd 2555 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( `' F " ( F
" U ) )  e.  J )
73kqffn 20094 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
8 dffn4 5807 . . . . . 6  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
97, 8sylib 196 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  F : X -onto-> ran  F )
109adantr 465 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  F : X -onto-> ran  F )
11 elqtop3 20072 . . . 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 470 . . 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 920 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( F " U )  e.  ( J qTop  F ) )
143kqval 20095 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  =  ( J qTop 
F ) )
1514adantr 465 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  (KQ `  J )  =  ( J qTop  F ) )
1613, 15eleqtrrd 2558 1  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( F " U )  e.  (KQ `  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2821    C_ wss 3481    |-> cmpt 4511   `'ccnv 5004   ran crn 5006   "cima 5008    Fn wfn 5589   -onto->wfo 5592   ` cfv 5594  (class class class)co 6295   qTop cqtop 14775  TopOnctopon 19264  KQckq 20062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-qtop 14779  df-topon 19271  df-kq 20063
This theorem is referenced by:  kqt0lem  20105  isr0  20106  regr1lem  20108  kqreglem1  20110  kqreglem2  20111  kqnrmlem1  20112  kqnrmlem2  20113
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