Theorem ist0-4 24103

Description: The topological indistinguishability map is injective iff the space is T₀. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

Ref	Expression
kqval.2	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})

Assertion

Ref	Expression
ist0-4	⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹:𝑋⟶1-1V))

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

Allowed substitution hints:   𝐹(𝑥, 𝑦)

Proof of Theorem ist0-4

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		kqval.2	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
2	1	kqfeq 24098	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦)))
3	2	3expb 1138	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦)))
4	3	imbi1d 344	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤) ↔ (∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦) → 𝑧 = 𝑤)))
5	4	2ralbidva 3228	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦) → 𝑧 = 𝑤)))
6	1	kqffn 24099	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
7		dffn2 6741	. . . 4 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋⟶V)
8	6, 7	sylib 221	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹:𝑋⟶V)
9		dff13 7289	. . . 4 ⊢ (𝐹:𝑋⟶1-1V ↔ (𝐹:𝑋⟶V ∧ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)))
10	9	baib 545	. . 3 ⊢ (𝐹:𝑋⟶V → (𝐹:𝑋⟶1-1V ↔ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)))
11	8, 10	syl 18	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐹:𝑋⟶1-1V ↔ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)))
12		ist0-2 23717	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦) → 𝑧 = 𝑤)))
13	5, 11, 12	3bitr4rd 315	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹:𝑋⟶1-1V))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416  Vcvv 3454   ↦ cmpt 5192   Fn wfn 6561  ⟶wf 6562  ⟶1-1wf1 6563  ‘cfv 6566  TopOnctopon 23283  Kol2ct0 23679

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-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-rab 3417  df-v 3456  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-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-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fv 6574  df-topon 23284  df-t0 23686

This theorem is used by:  t0kq  24192