Theorem t0kq 24192

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 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})

Assertion

Ref	Expression
t0kq	⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹 ∈ (𝐽Homeo(KQ‘𝐽))))

Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝑋,𝑦

Allowed substitution hints:   𝐹(𝑥, 𝑦)

Proof of Theorem t0kq

Step	Hyp	Ref	Expression
1		simpl 488	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → 𝐽 ∈ (TopOn‘𝑋))
2		t0kq.1	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
3	2	ist0-4 24103	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹:𝑋⟶1-1V))
4	3	biimpa 482	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → 𝐹:𝑋⟶1-1V)
5	1, 4	qtopf1 24190	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → 𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)))
6	2	kqval 24100	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
7	6	adantr 486	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
8	7	oveq2d 7467	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → (𝐽Homeo(KQ‘𝐽)) = (𝐽Homeo(𝐽 qTop 𝐹)))
9	5, 8	eleqtrrd 2867	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → 𝐹 ∈ (𝐽Homeo(KQ‘𝐽)))
10		hmphi 24151	. . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo(KQ‘𝐽)) → 𝐽 ≃ (KQ‘𝐽))
11		hmphsym 24156	. . . . 5 ⊢ (𝐽 ≃ (KQ‘𝐽) → (KQ‘𝐽) ≃ 𝐽)
12	10, 11	syl 18	. . . 4 ⊢ (𝐹 ∈ (𝐽Homeo(KQ‘𝐽)) → (KQ‘𝐽) ≃ 𝐽)
13	2	kqt0lem 24110	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ Kol2)
14		t0hmph 24164	. . . 4 ⊢ ((KQ‘𝐽) ≃ 𝐽 → ((KQ‘𝐽) ∈ Kol2 → 𝐽 ∈ Kol2))
15	12, 13, 14	syl2im 41	. . 3 ⊢ (𝐹 ∈ (𝐽Homeo(KQ‘𝐽)) → (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Kol2))
16	15	impcom 413	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (𝐽Homeo(KQ‘𝐽))) → 𝐽 ∈ Kol2)
17	9, 16	impbida 813	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹 ∈ (𝐽Homeo(KQ‘𝐽))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {crab 3416  Vcvv 3454   class class class wbr 5108   ↦ cmpt 5192  ⟶1-1wf1 6563  ‘cfv 6566  (class class class)co 7451   qTop cqtop 17725  TopOnctopon 23283  Kol2ct0 23679  KQckq 24067  Homeochmeo 24127   ≃ chmph 24128

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-1o 8514  df-map 8887  df-qtop 17729  df-top 23267  df-topon 23284  df-cn 23600  df-t0 23686  df-kq 24068  df-hmeo 24129  df-hmph 24130

This theorem is used by:  kqhmph  24193