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Theorem kqcld 17532
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 5107 . . . 4  |-  ( F
" U )  C_  ran  F
21a1i 10 . . 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 17530 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( `' F " ( F
" U ) )  =  U )
5 simpr 447 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  U  e.  ( Clsd `  J
) )
64, 5eqeltrd 2432 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( `' F " ( F
" U ) )  e.  ( Clsd `  J
) )
73kqffn 17522 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
8 dffn4 5540 . . . . . 6  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
97, 8sylib 188 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  F : X -onto-> ran  F )
10 qtopcld 17510 . . . . 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 649 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  ( ( F " U )  e.  ( Clsd `  ( J qTop  F ) )  <->  ( ( F " U )  C_  ran  F  /\  ( `' F " ( F
" U ) )  e.  ( Clsd `  J
) ) ) )
1211adantr 451 . . 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 888 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( F " U )  e.  ( Clsd `  ( J qTop  F ) ) )
143kqval 17523 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  =  ( J qTop 
F ) )
1514adantr 451 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  (KQ `  J )  =  ( J qTop  F ) )
1615fveq2d 5612 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  U  e.  ( Clsd `  J
) )  ->  ( Clsd `  (KQ `  J
) )  =  (
Clsd `  ( J qTop  F ) ) )
1713, 16eleqtrrd 2435 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 176    /\ wa 358    = wceq 1642    e. wcel 1710   {crab 2623    C_ wss 3228    e. cmpt 4158   `'ccnv 4770   ran crn 4772   "cima 4774    Fn wfn 5332   -onto->wfo 5335   ` cfv 5337  (class class class)co 5945   qTop cqtop 13505  TopOnctopon 16738   Clsdccld 16859  KQckq 17490
This theorem is referenced by:  kqreglem1  17538  kqnrmlem1  17540  kqnrmlem2  17541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-qtop 13509  df-top 16742  df-topon 16745  df-cld 16862  df-kq 17491
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