Theorem kqhmph 22970

Description: A topological space is T₀ iff it is homeomorphic to its Kolmogorov quotient. (Contributed by Mario Carneiro, 25-Aug-2015.)

Assertion

Ref	Expression
kqhmph	⊢ (𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽))

Proof of Theorem kqhmph

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		t0top 22480	. . . . . 6 ⊢ (𝐽 ∈ Kol2 → 𝐽 ∈ Top)
2		toptopon2 22067	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
3	1, 2	sylib 217	. . . . 5 ⊢ (𝐽 ∈ Kol2 → 𝐽 ∈ (TopOn‘∪ 𝐽))
4		eqid 2738	. . . . . 6 ⊢ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) = (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
5	4	t0kq 22969	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (𝐽 ∈ Kol2 ↔ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ∈ (𝐽Homeo(KQ‘𝐽))))
6	3, 5	syl 17	. . . 4 ⊢ (𝐽 ∈ Kol2 → (𝐽 ∈ Kol2 ↔ (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ∈ (𝐽Homeo(KQ‘𝐽))))
7	6	ibi 266	. . 3 ⊢ (𝐽 ∈ Kol2 → (𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ∈ (𝐽Homeo(KQ‘𝐽)))
8		hmphi 22928	. . 3 ⊢ ((𝑥 ∈ ∪ 𝐽 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) ∈ (𝐽Homeo(KQ‘𝐽)) → 𝐽 ≃ (KQ‘𝐽))
9	7, 8	syl 17	. 2 ⊢ (𝐽 ∈ Kol2 → 𝐽 ≃ (KQ‘𝐽))
10		hmphsym 22933	. . 3 ⊢ (𝐽 ≃ (KQ‘𝐽) → (KQ‘𝐽) ≃ 𝐽)
11		hmphtop1 22930	. . . 4 ⊢ (𝐽 ≃ (KQ‘𝐽) → 𝐽 ∈ Top)
12		kqt0 22897	. . . 4 ⊢ (𝐽 ∈ Top ↔ (KQ‘𝐽) ∈ Kol2)
13	11, 12	sylib 217	. . 3 ⊢ (𝐽 ≃ (KQ‘𝐽) → (KQ‘𝐽) ∈ Kol2)
14		t0hmph 22941	. . 3 ⊢ ((KQ‘𝐽) ≃ 𝐽 → ((KQ‘𝐽) ∈ Kol2 → 𝐽 ∈ Kol2))
15	10, 13, 14	sylc 65	. 2 ⊢ (𝐽 ≃ (KQ‘𝐽) → 𝐽 ∈ Kol2)
16	9, 15	impbii 208	1 ⊢ (𝐽 ∈ Kol2 ↔ 𝐽 ≃ (KQ‘𝐽))

Syntax hints:   ↔ wb 205   ∈ wcel 2106  {crab 3068  ∪ cuni 4839   class class class wbr 5074   ↦ cmpt 5157  ‘cfv 6433  (class class class)co 7275  Topctop 22042  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:  ist1-5lem  22971  t1r0  22972