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Theorem kqcld 19971
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 5346 . . . 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 19969 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( `' F " ( F
" U ) )  =  U )
5 simpr 461 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  U  e.  ( Clsd `  J
) )
64, 5eqeltrd 2555 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( `' F " ( F
" U ) )  e.  ( Clsd `  J
) )
73kqffn 19961 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
8 dffn4 5799 . . . . . 6  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
97, 8sylib 196 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  F : X -onto-> ran  F )
10 qtopcld 19949 . . . . 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 668 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( ( F " U )  e.  ( Clsd `  ( J qTop  F ) )  <->  ( ( F " U )  C_  ran  F  /\  ( `' F " ( F
" U ) )  e.  ( Clsd `  J
) ) ) )
1211adantr 465 . . 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 19962 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  =  ( J qTop 
F ) )
1514adantr 465 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  (KQ `  J )  =  ( J qTop  F ) )
1615fveq2d 5868 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( Clsd `  (KQ `  J
) )  =  (
Clsd `  ( J qTop  F ) ) )
1713, 16eleqtrrd 2558 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 369    = wceq 1379    e. wcel 1767   {crab 2818    C_ wss 3476    |-> cmpt 4505   `'ccnv 4998   ran crn 5000   "cima 5002    Fn wfn 5581   -onto->wfo 5584   ` cfv 5586  (class class class)co 6282   qTop cqtop 14754  TopOnctopon 19162   Clsdccld 19283  KQckq 19929
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-qtop 14758  df-top 19166  df-topon 19169  df-cld 19286  df-kq 19930
This theorem is referenced by:  kqreglem1  19977  kqnrmlem1  19979  kqnrmlem2  19980
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