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Theorem kqcld 20321
Description: The topological indistinguishability map is a closed 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
kqcld  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( F " U )  e.  ( Clsd `  (KQ `  J ) ) )
Distinct variable groups:    x, y, J    x, X, y
Allowed substitution hints:    U( x, y)    F( x, y)

Proof of Theorem kqcld
StepHypRef Expression
1 imassrn 5260 . . . 4  |-  ( F
" U )  C_  ran  F
21a1i 11 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( F " U )  C_  ran  F )
3 kqval.2 . . . . 5  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
43kqcldsat 20319 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( `' F " ( F
" U ) )  =  U )
5 simpr 459 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  U  e.  ( Clsd `  J
) )
64, 5eqeltrd 2470 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( `' F " ( F
" U ) )  e.  ( Clsd `  J
) )
73kqffn 20311 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
8 dffn4 5709 . . . . . 6  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
97, 8sylib 196 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  F : X -onto-> ran  F )
10 qtopcld 20299 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  F : X -onto-> ran  F )  -> 
( ( F " U )  e.  (
Clsd `  ( J qTop  F ) )  <->  ( ( F " U )  C_  ran  F  /\  ( `' F " ( F
" U ) )  e.  ( Clsd `  J
) ) ) )
119, 10mpdan 666 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( ( F " U )  e.  ( Clsd `  ( J qTop  F ) )  <->  ( ( F " U )  C_  ran  F  /\  ( `' F " ( F
" U ) )  e.  ( Clsd `  J
) ) ) )
1211adantr 463 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  (
( F " U
)  e.  ( Clsd `  ( J qTop  F ) )  <->  ( ( F
" U )  C_  ran  F  /\  ( `' F " ( F
" U ) )  e.  ( Clsd `  J
) ) ) )
132, 6, 12mpbir2and 920 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( F " U )  e.  ( Clsd `  ( J qTop  F ) ) )
143kqval 20312 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  =  ( J qTop 
F ) )
1514adantr 463 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  (KQ `  J )  =  ( J qTop  F ) )
1615fveq2d 5778 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( Clsd `  (KQ `  J
) )  =  (
Clsd `  ( J qTop  F ) ) )
1713, 16eleqtrrd 2473 1  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( F " U )  e.  ( Clsd `  (KQ `  J ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   {crab 2736    C_ wss 3389    |-> cmpt 4425   `'ccnv 4912   ran crn 4914   "cima 4916    Fn wfn 5491   -onto->wfo 5494   ` cfv 5496  (class class class)co 6196   qTop cqtop 14910  TopOnctopon 19480   Clsdccld 19602  KQckq 20279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-qtop 14914  df-top 19484  df-topon 19487  df-cld 19605  df-kq 20280
This theorem is referenced by:  kqreglem1  20327  kqnrmlem1  20329  kqnrmlem2  20330
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