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Theorem t0kq 23322
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 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
Assertion
Ref Expression
t0kq (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ Kol2 ↔ 𝐹 ∈ (𝐽Homeo(KQβ€˜π½))))
Distinct variable groups:   π‘₯,𝑦,𝐽   π‘₯,𝑋,𝑦
Allowed substitution hints:   𝐹(π‘₯,𝑦)

Proof of Theorem t0kq
StepHypRef Expression
1 simpl 484 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 ∈ Kol2) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
2 t0kq.1 . . . . . 6 𝐹 = (π‘₯ ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ π‘₯ ∈ 𝑦})
32ist0-4 23233 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ Kol2 ↔ 𝐹:𝑋–1-1β†’V))
43biimpa 478 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 ∈ Kol2) β†’ 𝐹:𝑋–1-1β†’V)
51, 4qtopf1 23320 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 ∈ Kol2) β†’ 𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)))
62kqval 23230 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (KQβ€˜π½) = (𝐽 qTop 𝐹))
76adantr 482 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 ∈ Kol2) β†’ (KQβ€˜π½) = (𝐽 qTop 𝐹))
87oveq2d 7425 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 ∈ Kol2) β†’ (𝐽Homeo(KQβ€˜π½)) = (𝐽Homeo(𝐽 qTop 𝐹)))
95, 8eleqtrrd 2837 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐽 ∈ Kol2) β†’ 𝐹 ∈ (𝐽Homeo(KQβ€˜π½)))
10 hmphi 23281 . . . . 5 (𝐹 ∈ (𝐽Homeo(KQβ€˜π½)) β†’ 𝐽 ≃ (KQβ€˜π½))
11 hmphsym 23286 . . . . 5 (𝐽 ≃ (KQβ€˜π½) β†’ (KQβ€˜π½) ≃ 𝐽)
1210, 11syl 17 . . . 4 (𝐹 ∈ (𝐽Homeo(KQβ€˜π½)) β†’ (KQβ€˜π½) ≃ 𝐽)
132kqt0lem 23240 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (KQβ€˜π½) ∈ Kol2)
14 t0hmph 23294 . . . 4 ((KQβ€˜π½) ≃ 𝐽 β†’ ((KQβ€˜π½) ∈ Kol2 β†’ 𝐽 ∈ Kol2))
1512, 13, 14syl2im 40 . . 3 (𝐹 ∈ (𝐽Homeo(KQβ€˜π½)) β†’ (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Kol2))
1615impcom 409 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (𝐽Homeo(KQβ€˜π½))) β†’ 𝐽 ∈ Kol2)
179, 16impbida 800 1 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ Kol2 ↔ 𝐹 ∈ (𝐽Homeo(KQβ€˜π½))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {crab 3433  Vcvv 3475   class class class wbr 5149   ↦ cmpt 5232  β€“1-1β†’wf1 6541  β€˜cfv 6544  (class class class)co 7409   qTop cqtop 17449  TopOnctopon 22412  Kol2ct0 22810  KQckq 23197  Homeochmeo 23257   ≃ chmph 23258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-1o 8466  df-map 8822  df-qtop 17453  df-top 22396  df-topon 22413  df-cn 22731  df-t0 22817  df-kq 23198  df-hmeo 23259  df-hmph 23260
This theorem is referenced by:  kqhmph  23323
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