MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  kqopn Unicode version

Theorem kqopn 17425
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 5025 . . . 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 17422 . . . 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 2357 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  J )  ->  ( `' F " ( F
" U ) )  e.  J )
73kqffn 17416 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
8 dffn4 5457 . . . . . 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 17394 . . . 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 17417 . . 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 2360 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 1623    e. wcel 1684   {crab 2547    C_ wss 3152    e. cmpt 4077   `'ccnv 4688   ran crn 4690   "cima 4692    Fn wfn 5250   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   qTop cqtop 13406  TopOnctopon 16632  KQckq 17384
This theorem is referenced by:  kqt0lem  17427  isr0  17428  regr1lem  17430  kqreglem1  17432  kqreglem2  17433  kqnrmlem1  17434  kqnrmlem2  17435
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-qtop 13410  df-topon 16639  df-kq 17385
  Copyright terms: Public domain W3C validator