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Theorem kqcld 17724
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 5179 . . . 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 17722 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( `' F " ( F
" U ) )  =  U )
5 simpr 448 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  U  e.  ( Clsd `  J
) )
64, 5eqeltrd 2482 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( `' F " ( F
" U ) )  e.  ( Clsd `  J
) )
73kqffn 17714 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
8 dffn4 5622 . . . . . 6  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
97, 8sylib 189 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  F : X -onto-> ran  F )
10 qtopcld 17702 . . . . 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 650 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( ( F " U )  e.  ( Clsd `  ( J qTop  F ) )  <->  ( ( F " U )  C_  ran  F  /\  ( `' F " ( F
" U ) )  e.  ( Clsd `  J
) ) ) )
1211adantr 452 . . 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 889 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( F " U )  e.  ( Clsd `  ( J qTop  F ) ) )
143kqval 17715 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  =  ( J qTop 
F ) )
1514adantr 452 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  (KQ `  J )  =  ( J qTop  F ) )
1615fveq2d 5695 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( Clsd `  (KQ `  J
) )  =  (
Clsd `  ( J qTop  F ) ) )
1713, 16eleqtrrd 2485 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   {crab 2674    C_ wss 3284    e. cmpt 4230   `'ccnv 4840   ran crn 4842   "cima 4844    Fn wfn 5412   -onto->wfo 5415   ` cfv 5417  (class class class)co 6044   qTop cqtop 13688  TopOnctopon 16918   Clsdccld 17039  KQckq 17682
This theorem is referenced by:  kqreglem1  17730  kqnrmlem1  17732  kqnrmlem2  17733
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-qtop 13692  df-top 16922  df-topon 16925  df-cld 17042  df-kq 17683
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