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Theorem kqtopon 17751
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 17750 . 2  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  =  ( J qTop 
F ) )
31kqffn 17749 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
4 dffn4 5651 . . . 4  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
53, 4sylib 189 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  F : X -onto-> ran  F )
6 qtoptopon 17728 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F : X -onto-> ran  F )  -> 
( J qTop  F )  e.  (TopOn `  ran  F ) )
75, 6mpdan 650 . 2  |-  ( J  e.  (TopOn `  X
)  ->  ( J qTop  F )  e.  (TopOn `  ran  F ) )
82, 7eqeltrd 2509 1  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  e.  (TopOn `  ran  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {crab 2701    e. cmpt 4258   ran crn 4871    Fn wfn 5441   -onto->wfo 5444   ` cfv 5446  (class class class)co 6073   qTop cqtop 13721  TopOnctopon 16951  KQckq 17717
This theorem is referenced by:  kqt0lem  17760  isr0  17761  r0cld  17762  regr1lem2  17764  kqreglem1  17765  kqreglem2  17766  kqnrmlem1  17767  kqnrmlem2  17768  kqtop  17769
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-qtop 13725  df-top 16955  df-topon 16958  df-kq 17718
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