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Theorem kqid 17760
Description: The topological indistinguishability map is a continuous function into the Kolmogorov quotient. (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
kqid  |-  ( J  e.  (TopOn `  X
)  ->  F  e.  ( J  Cn  (KQ `  J ) ) )
Distinct variable groups:    x, y, J    x, X, y
Allowed substitution hints:    F( x, y)

Proof of Theorem kqid
StepHypRef Expression
1 kqval.2 . . . 4  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
21kqffn 17757 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  F  Fn  X )
3 qtopid 17737 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  Fn  X )  ->  F  e.  ( J  Cn  ( J qTop  F ) ) )
42, 3mpdan 650 . 2  |-  ( J  e.  (TopOn `  X
)  ->  F  e.  ( J  Cn  ( J qTop  F ) ) )
51kqval 17758 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  =  ( J qTop 
F ) )
65oveq2d 6097 . 2  |-  ( J  e.  (TopOn `  X
)  ->  ( J  Cn  (KQ `  J ) )  =  ( J  Cn  ( J qTop  F
) ) )
74, 6eleqtrrd 2513 1  |-  ( J  e.  (TopOn `  X
)  ->  F  e.  ( J  Cn  (KQ `  J ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {crab 2709    e. cmpt 4266    Fn wfn 5449   ` cfv 5454  (class class class)co 6081   qTop cqtop 13729  TopOnctopon 16959    Cn ccn 17288  KQckq 17725
This theorem is referenced by:  isr0  17769  r0cld  17770  kqreglem1  17773  kqreglem2  17774  kqnrmlem1  17775  kqnrmlem2  17776
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-qtop 13733  df-top 16963  df-topon 16966  df-cn 17291  df-kq 17726
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