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Theorem kqcld 17772
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 5219 . . . 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 17770 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( `' F " ( F
" U ) )  =  U )
5 simpr 449 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  U  e.  ( Clsd `  J
) )
64, 5eqeltrd 2512 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( `' F " ( F
" U ) )  e.  ( Clsd `  J
) )
73kqffn 17762 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
8 dffn4 5662 . . . . . 6  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
97, 8sylib 190 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  F : X -onto-> ran  F )
10 qtopcld 17750 . . . . 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 651 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( ( F " U )  e.  ( Clsd `  ( J qTop  F ) )  <->  ( ( F " U )  C_  ran  F  /\  ( `' F " ( F
" U ) )  e.  ( Clsd `  J
) ) ) )
1211adantr 453 . . 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 890 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( F " U )  e.  ( Clsd `  ( J qTop  F ) ) )
143kqval 17763 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  =  ( J qTop 
F ) )
1514adantr 453 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  (KQ `  J )  =  ( J qTop  F ) )
1615fveq2d 5735 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( Clsd `  (KQ `  J
) )  =  (
Clsd `  ( J qTop  F ) ) )
1713, 16eleqtrrd 2515 1  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( F " U )  e.  ( Clsd `  (KQ `  J ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2711    C_ wss 3322    e. cmpt 4269   `'ccnv 4880   ran crn 4882   "cima 4884    Fn wfn 5452   -onto->wfo 5455   ` cfv 5457  (class class class)co 6084   qTop cqtop 13734  TopOnctopon 16964   Clsdccld 17085  KQckq 17730
This theorem is referenced by:  kqreglem1  17778  kqnrmlem1  17780  kqnrmlem2  17781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-qtop 13738  df-top 16968  df-topon 16971  df-cld 17088  df-kq 17731
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