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Theorem ist0-4 22343
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 22338 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑦𝐽 (𝑧𝑦𝑤𝑦)))
323expb 1117 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑦𝐽 (𝑧𝑦𝑤𝑦)))
43imbi1d 345 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧𝑋𝑤𝑋)) → (((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤) ↔ (∀𝑦𝐽 (𝑧𝑦𝑤𝑦) → 𝑧 = 𝑤)))
542ralbidva 3193 . 2 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑧𝑋𝑤𝑋 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤) ↔ ∀𝑧𝑋𝑤𝑋 (∀𝑦𝐽 (𝑧𝑦𝑤𝑦) → 𝑧 = 𝑤)))
61kqffn 22339 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
7 dffn2 6507 . . . 4 (𝐹 Fn 𝑋𝐹:𝑋⟶V)
86, 7sylib 221 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐹:𝑋⟶V)
9 dff13 7007 . . . 4 (𝐹:𝑋1-1→V ↔ (𝐹:𝑋⟶V ∧ ∀𝑧𝑋𝑤𝑋 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
109baib 539 . . 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 21958 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ ∀𝑧𝑋𝑤𝑋 (∀𝑦𝐽 (𝑧𝑦𝑤𝑦) → 𝑧 = 𝑤)))
135, 11, 123bitr4rd 315 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹:𝑋1-1→V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3133  {crab 3137  Vcvv 3480  cmpt 5133   Fn wfn 6340  wf 6341  1-1wf1 6342  cfv 6345  TopOnctopon 21524  Kol2ct0 21920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fv 6353  df-topon 21525  df-t0 21927
This theorem is referenced by:  t0kq  22432
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