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Theorem ist0-4 21442
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 21437 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑦𝐽 (𝑧𝑦𝑤𝑦)))
323expb 1263 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑦𝐽 (𝑧𝑦𝑤𝑦)))
43imbi1d 331 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧𝑋𝑤𝑋)) → (((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤) ↔ (∀𝑦𝐽 (𝑧𝑦𝑤𝑦) → 𝑧 = 𝑤)))
542ralbidva 2982 . 2 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑧𝑋𝑤𝑋 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤) ↔ ∀𝑧𝑋𝑤𝑋 (∀𝑦𝐽 (𝑧𝑦𝑤𝑦) → 𝑧 = 𝑤)))
61kqffn 21438 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
7 dffn2 6004 . . . 4 (𝐹 Fn 𝑋𝐹:𝑋⟶V)
86, 7sylib 208 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐹:𝑋⟶V)
9 dff13 6466 . . . 4 (𝐹:𝑋1-1→V ↔ (𝐹:𝑋⟶V ∧ ∀𝑧𝑋𝑤𝑋 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
109baib 943 . . 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 21058 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ ∀𝑧𝑋𝑤𝑋 (∀𝑦𝐽 (𝑧𝑦𝑤𝑦) → 𝑧 = 𝑤)))
135, 11, 123bitr4rd 301 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹:𝑋1-1→V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3186  cmpt 4673   Fn wfn 5842  wf 5843  1-1wf1 5844  cfv 5847  TopOnctopon 20618  Kol2ct0 21020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fv 5855  df-topon 20623  df-t0 21027
This theorem is referenced by:  t0kq  21531
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