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Theorem ist0-4 23721
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 23716 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝑋𝑤𝑋) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑦𝐽 (𝑧𝑦𝑤𝑦)))
323expb 1117 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧𝑋𝑤𝑋)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ∀𝑦𝐽 (𝑧𝑦𝑤𝑦)))
43imbi1d 340 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧𝑋𝑤𝑋)) → (((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤) ↔ (∀𝑦𝐽 (𝑧𝑦𝑤𝑦) → 𝑧 = 𝑤)))
542ralbidva 3207 . 2 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑧𝑋𝑤𝑋 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤) ↔ ∀𝑧𝑋𝑤𝑋 (∀𝑦𝐽 (𝑧𝑦𝑤𝑦) → 𝑧 = 𝑤)))
61kqffn 23717 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
7 dffn2 6722 . . . 4 (𝐹 Fn 𝑋𝐹:𝑋⟶V)
86, 7sylib 217 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐹:𝑋⟶V)
9 dff13 7262 . . . 4 (𝐹:𝑋1-1→V ↔ (𝐹:𝑋⟶V ∧ ∀𝑧𝑋𝑤𝑋 ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
109baib 534 . . 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 23336 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ ∀𝑧𝑋𝑤𝑋 (∀𝑦𝐽 (𝑧𝑦𝑤𝑦) → 𝑧 = 𝑤)))
135, 11, 123bitr4rd 311 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹:𝑋1-1→V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wral 3051  {crab 3419  Vcvv 3462  cmpt 5228   Fn wfn 6541  wf 6542  1-1wf1 6543  cfv 6546  TopOnctopon 22900  Kol2ct0 23298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fv 6554  df-topon 22901  df-t0 23305
This theorem is referenced by:  t0kq  23810
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