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Theorem kqtopon 20673
Description: The Kolmogorov quotient is a topology on the quotient set. (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
kqtopon  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  e.  (TopOn `  ran  F ) )
Distinct variable groups:    x, y, J    x, X, y
Allowed substitution hints:    F( x, y)

Proof of Theorem kqtopon
StepHypRef Expression
1 kqval.2 . . 3  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
21kqval 20672 . 2  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  =  ( J qTop 
F ) )
31kqffn 20671 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
4 dffn4 5816 . . . 4  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
53, 4sylib 199 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  F : X -onto-> ran  F )
6 qtoptopon 20650 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F : X -onto-> ran  F )  -> 
( J qTop  F )  e.  (TopOn `  ran  F ) )
75, 6mpdan 672 . 2  |-  ( J  e.  (TopOn `  X
)  ->  ( J qTop  F )  e.  (TopOn `  ran  F ) )
82, 7eqeltrd 2517 1  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  e.  (TopOn `  ran  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870   {crab 2786    |-> cmpt 4484   ran crn 4855    Fn wfn 5596   -onto->wfo 5599   ` cfv 5601  (class class class)co 6305   qTop cqtop 15360  TopOnctopon 19849  KQckq 20639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-qtop 15364  df-top 19852  df-topon 19854  df-kq 20640
This theorem is referenced by:  kqt0lem  20682  isr0  20683  r0cld  20684  regr1lem2  20686  kqreglem1  20687  kqreglem2  20688  kqnrmlem1  20689  kqnrmlem2  20690  kqtop  20691
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