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 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})

Assertion

Ref	Expression
ist0-4	⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹:𝑋–1-1→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.

Step	Hyp	Ref	Expression
1		kqval.2	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
2	1	kqfeq 23716	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦)))
3	2	3expb 1117	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦)))
4	3	imbi1d 340	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤) ↔ (∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦) → 𝑧 = 𝑤)))
5	4	2ralbidva 3207	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤) ↔ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦) → 𝑧 = 𝑤)))
6	1	kqffn 23717	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
7		dffn2 6722	. . . 4 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋⟶V)
8	6, 7	sylib 217	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹:𝑋⟶V)
9		dff13 7262	. . . 4 ⊢ (𝐹:𝑋–1-1→V ↔ (𝐹:𝑋⟶V ∧ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)))
10	9	baib 534	. . 3 ⊢ (𝐹:𝑋⟶V → (𝐹:𝑋–1-1→V ↔ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)))
11	8, 10	syl 17	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐹:𝑋–1-1→V ↔ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)))
12		ist0-2 23336	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦) → 𝑧 = 𝑤)))
13	5, 11, 12	3bitr4rd 311	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Kol2 ↔ 𝐹:𝑋–1-1→V))

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  –1-1→wf1 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