Theorem t0kq 22969

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 483	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → 𝐽 ∈ (TopOn‘𝑋))
2		t0kq.1	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
3	2	ist0-4 22880	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹:𝑋–1-1→V))
4	3	biimpa 477	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → 𝐹:𝑋–1-1→V)
5	1, 4	qtopf1 22967	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → 𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)))
6	2	kqval 22877	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
7	6	adantr 481	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → (KQ‘𝐽) = (𝐽 qTop 𝐹))
8	7	oveq2d 7291	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → (𝐽Homeo(KQ‘𝐽)) = (𝐽Homeo(𝐽 qTop 𝐹)))
9	5, 8	eleqtrrd 2842	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → 𝐹 ∈ (𝐽Homeo(KQ‘𝐽)))
10		hmphi 22928	. . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo(KQ‘𝐽)) → 𝐽 ≃ (KQ‘𝐽))
11		hmphsym 22933	. . . . 5 ⊢ (𝐽 ≃ (KQ‘𝐽) → (KQ‘𝐽) ≃ 𝐽)
12	10, 11	syl 17	. . . 4 ⊢ (𝐹 ∈ (𝐽Homeo(KQ‘𝐽)) → (KQ‘𝐽) ≃ 𝐽)
13	2	kqt0lem 22887	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ Kol2)
14		t0hmph 22941	. . . 4 ⊢ ((KQ‘𝐽) ≃ 𝐽 → ((KQ‘𝐽) ∈ Kol2 → 𝐽 ∈ Kol2))
15	12, 13, 14	syl2im 40	. . 3 ⊢ (𝐹 ∈ (𝐽Homeo(KQ‘𝐽)) → (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Kol2))
16	15	impcom 408	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (𝐽Homeo(KQ‘𝐽))) → 𝐽 ∈ Kol2)
17	9, 16	impbida 798	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹 ∈ (𝐽Homeo(KQ‘𝐽))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {crab 3068  Vcvv 3432   class class class wbr 5074   ↦ cmpt 5157  –1-1→wf1 6430  ‘cfv 6433  (class class class)co 7275   qTop cqtop 17214  TopOnctopon 22059  Kol2ct0 22457  KQckq 22844  Homeochmeo 22904   ≃ chmph 22905

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-1o 8297  df-map 8617  df-qtop 17218  df-top 22043  df-topon 22060  df-cn 22378  df-t0 22464  df-kq 22845  df-hmeo 22906  df-hmph 22907

This theorem is referenced by:  kqhmph  22970