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Theorem t0kq 20445
Description: A topological space is T0 iff the quotient map is a homeomorphism onto the space's Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
t0kq.1  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
Assertion
Ref Expression
t0kq  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e.  Kol2  <->  F  e.  ( J Homeo (KQ `  J
) ) ) )
Distinct variable groups:    x, y, J    x, X, y
Allowed substitution hints:    F( x, y)

Proof of Theorem t0kq
StepHypRef Expression
1 simpl 457 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  J  e.  Kol2 )  ->  J  e.  (TopOn `  X )
)
2 t0kq.1 . . . . . 6  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
32ist0-4 20356 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e.  Kol2  <->  F : X -1-1-> _V ) )
43biimpa 484 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  J  e.  Kol2 )  ->  F : X -1-1-> _V )
51, 4qtopf1 20443 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  J  e.  Kol2 )  ->  F  e.  ( J Homeo ( J qTop 
F ) ) )
62kqval 20353 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  =  ( J qTop 
F ) )
76adantr 465 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  J  e.  Kol2 )  ->  (KQ `  J )  =  ( J qTop  F ) )
87oveq2d 6312 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  J  e.  Kol2 )  ->  ( J Homeo (KQ `  J
) )  =  ( J Homeo ( J qTop  F
) ) )
95, 8eleqtrrd 2548 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  J  e.  Kol2 )  ->  F  e.  ( J Homeo (KQ `  J ) ) )
10 hmphi 20404 . . . . 5  |-  ( F  e.  ( J Homeo (KQ
`  J ) )  ->  J  ~=  (KQ `  J ) )
11 hmphsym 20409 . . . . 5  |-  ( J  ~=  (KQ `  J
)  ->  (KQ `  J
)  ~=  J )
1210, 11syl 16 . . . 4  |-  ( F  e.  ( J Homeo (KQ
`  J ) )  ->  (KQ `  J
)  ~=  J )
132kqt0lem 20363 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  (KQ `  J
)  e.  Kol2 )
14 t0hmph 20417 . . . 4  |-  ( (KQ
`  J )  ~=  J  ->  ( (KQ `  J )  e.  Kol2  ->  J  e.  Kol2 ) )
1512, 13, 14syl2im 38 . . 3  |-  ( F  e.  ( J Homeo (KQ
`  J ) )  ->  ( J  e.  (TopOn `  X )  ->  J  e.  Kol2 )
)
1615impcom 430 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( J Homeo (KQ `  J ) ) )  ->  J  e.  Kol2 )
179, 16impbida 832 1  |-  ( J  e.  (TopOn `  X
)  ->  ( J  e.  Kol2  <->  F  e.  ( J Homeo (KQ `  J
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   {crab 2811   _Vcvv 3109   class class class wbr 4456    |-> cmpt 4515   -1-1->wf1 5591   ` cfv 5594  (class class class)co 6296   qTop cqtop 14920  TopOnctopon 19522   Kol2ct0 19934  KQckq 20320   Homeochmeo 20380    ~= chmph 20381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-1o 7148  df-map 7440  df-qtop 14924  df-top 19526  df-topon 19529  df-cn 19855  df-t0 19941  df-kq 20321  df-hmeo 20382  df-hmph 20383
This theorem is referenced by:  kqhmph  20446
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