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Theorem kqtopon 20055
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 20054 . 2  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  =  ( J qTop 
F ) )
31kqffn 20053 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
4 dffn4 5801 . . . 4  |-  ( F  Fn  X  <->  F : X -onto-> ran  F )
53, 4sylib 196 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  F : X -onto-> ran  F )
6 qtoptopon 20032 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F : X -onto-> ran  F )  -> 
( J qTop  F )  e.  (TopOn `  ran  F ) )
75, 6mpdan 668 . 2  |-  ( J  e.  (TopOn `  X
)  ->  ( J qTop  F )  e.  (TopOn `  ran  F ) )
82, 7eqeltrd 2555 1  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  e.  (TopOn `  ran  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   {crab 2818    |-> cmpt 4505   ran crn 5000    Fn wfn 5583   -onto->wfo 5586   ` cfv 5588  (class class class)co 6285   qTop cqtop 14761  TopOnctopon 19202  KQckq 20021
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 6577
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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-qtop 14765  df-top 19206  df-topon 19209  df-kq 20022
This theorem is referenced by:  kqt0lem  20064  isr0  20065  r0cld  20066  regr1lem2  20068  kqreglem1  20069  kqreglem2  20070  kqnrmlem1  20071  kqnrmlem2  20072  kqtop  20073
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