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Theorem gneispacess 44158
Description: All supersets of a neighborhood of a point (limited to the domain of the neighborhood space) are also neighborhoods of that point. (Contributed by RP, 15-Apr-2021.)
Hypothesis
Ref Expression
gneispace.a 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}
Assertion
Ref Expression
gneispacess (𝐹𝐴 → ∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)))
Distinct variable groups:   𝑛,𝐹,𝑝,𝑓   𝐹,𝑠,𝑓   𝑓,𝑛,𝑝
Allowed substitution hints:   𝐴(𝑓,𝑛,𝑠,𝑝)

Proof of Theorem gneispacess
StepHypRef Expression
1 gneispace.a . . . 4 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}
21gneispace2 44145 . . 3 (𝐹𝐴 → (𝐹𝐴 ↔ (𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝))))))
32ibi 267 . 2 (𝐹𝐴 → (𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)))))
4 simpr 484 . . 3 ((𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝))) → ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)))
542ralimi 3123 . 2 (∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝))) → ∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)))
63, 5simpl2im 503 1 (𝐹𝐴 → ∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {cab 2714  wral 3061  cdif 3948  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626  dom cdm 5685  wf 6557  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569
This theorem is referenced by:  gneispacess2  44159
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