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Theorem ist0-4 21342
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 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
Assertion
Ref Expression
ist0-4 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹:𝑋1-1→V))
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.
StepHypRef Expression
1 kqval.2 . . . . . 6 𝐹 = (𝑥𝑋 ↦ {𝑦𝐽𝑥𝑦})
21kqfeq 21337 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑦𝐽 (𝑧𝑦𝑤𝑦)))
323expb 1258 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑦𝐽 (𝑧𝑦𝑤𝑦)))
43imbi1d 330 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧𝑋𝑤𝑋)) → (((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤) ↔ (∀𝑦𝐽 (𝑧𝑦𝑤𝑦) → 𝑧 = 𝑤)))
542ralbidva 2971 . 2 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑧𝑋𝑤𝑋 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤) ↔ ∀𝑧𝑋𝑤𝑋 (∀𝑦𝐽 (𝑧𝑦𝑤𝑦) → 𝑧 = 𝑤)))
61kqffn 21338 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
7 dffn2 5960 . . . 4 (𝐹 Fn 𝑋𝐹:𝑋⟶V)
86, 7sylib 207 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐹:𝑋⟶V)
9 dff13 6416 . . . 4 (𝐹:𝑋1-1→V ↔ (𝐹:𝑋⟶V ∧ ∀𝑧𝑋𝑤𝑋 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
109baib 942 . . 3 (𝐹:𝑋⟶V → (𝐹:𝑋1-1→V ↔ ∀𝑧𝑋𝑤𝑋 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
118, 10syl 17 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐹:𝑋1-1→V ↔ ∀𝑧𝑋𝑤𝑋 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
12 ist0-2 20958 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ ∀𝑧𝑋𝑤𝑋 (∀𝑦𝐽 (𝑧𝑦𝑤𝑦) → 𝑧 = 𝑤)))
135, 11, 123bitr4rd 300 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹:𝑋1-1→V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  {crab 2900  Vcvv 3173  cmpt 4643   Fn wfn 5799  wf 5800  1-1wf1 5801  cfv 5804  TopOnctopon 20518  Kol2ct0 20920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fv 5812  df-topon 20523  df-t0 20927
This theorem is referenced by:  t0kq  21431
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