Theorem t0kq 24149

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 24060	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹:𝑋–1-1→V))
4	3	biimpa 482	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → 𝐹:𝑋–1-1→V)
5	1, 4	qtopf1 24147	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → 𝐹 ∈ (𝐽Homeo(𝐽 qTop 𝐹)))
6	2	kqval 24057	. . . . 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 7443	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → (𝐽Homeo(KQ‘𝐽)) = (𝐽Homeo(𝐽 qTop 𝐹)))
9	5, 8	eleqtrrd 2866	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Kol2) → 𝐹 ∈ (𝐽Homeo(KQ‘𝐽)))
10		hmphi 24108	. . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo(KQ‘𝐽)) → 𝐽 ≃ (KQ‘𝐽))
11		hmphsym 24113	. . . . 5 ⊢ (𝐽 ≃ (KQ‘𝐽) → (KQ‘𝐽) ≃ 𝐽)
12	10, 11	syl 18	. . . 4 ⊢ (𝐹 ∈ (𝐽Homeo(KQ‘𝐽)) → (KQ‘𝐽) ≃ 𝐽)
13	2	kqt0lem 24067	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ Kol2)
14		t0hmph 24121	. . . 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 1569   ∈ wcel 2146  {crab 3415  Vcvv 3453   class class class wbr 5105   ↦ cmpt 5188  –1-1→wf1 6543  ‘cfv 6546  (class class class)co 7427   qTop cqtop 17687  TopOnctopon 23240  Kol2ct0 23636  KQckq 24024  Homeochmeo 24084   ≃ chmph 24085

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-1st 8008  df-2nd 8009  df-1o 8487  df-map 8860  df-qtop 17691  df-top 23224  df-topon 23241  df-cn 23557  df-t0 23643  df-kq 24025  df-hmeo 24086  df-hmph 24087

This theorem is used by:  kqhmph  24150