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Theorem ist0-4 23738
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 23733 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑦𝐽 (𝑧𝑦𝑤𝑦)))
323expb 1120 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑦𝐽 (𝑧𝑦𝑤𝑦)))
43imbi1d 341 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧𝑋𝑤𝑋)) → (((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤) ↔ (∀𝑦𝐽 (𝑧𝑦𝑤𝑦) → 𝑧 = 𝑤)))
542ralbidva 3218 . 2 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑧𝑋𝑤𝑋 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤) ↔ ∀𝑧𝑋𝑤𝑋 (∀𝑦𝐽 (𝑧𝑦𝑤𝑦) → 𝑧 = 𝑤)))
61kqffn 23734 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
7 dffn2 6737 . . . 4 (𝐹 Fn 𝑋𝐹:𝑋⟶V)
86, 7sylib 218 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐹:𝑋⟶V)
9 dff13 7276 . . . 4 (𝐹:𝑋1-1→V ↔ (𝐹:𝑋⟶V ∧ ∀𝑧𝑋𝑤𝑋 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
109baib 535 . . 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 23353 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ ∀𝑧𝑋𝑤𝑋 (∀𝑦𝐽 (𝑧𝑦𝑤𝑦) → 𝑧 = 𝑤)))
135, 11, 123bitr4rd 312 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹:𝑋1-1→V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wral 3060  {crab 3435  Vcvv 3479  cmpt 5224   Fn wfn 6555  wf 6556  1-1wf1 6557  cfv 6560  TopOnctopon 22917  Kol2ct0 23315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fv 6568  df-topon 22918  df-t0 23322
This theorem is referenced by:  t0kq  23827
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