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Theorem gneispacess2 40768
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
gneispacess2 (((𝐹𝐴𝑃 ∈ dom 𝐹) ∧ (𝑁 ∈ (𝐹𝑃) ∧ 𝑆 ∈ 𝒫 dom 𝐹𝑁𝑆)) → 𝑆 ∈ (𝐹𝑃))
Distinct variable groups:   𝑛,𝐹,𝑝,𝑓,𝑠   𝑃,𝑝,𝑛   𝑛,𝑁   𝑆,𝑠   𝑛,𝑠,𝑁   𝑠,𝑝,𝑃
Allowed substitution hints:   𝐴(𝑓,𝑛,𝑠,𝑝)   𝑃(𝑓)   𝑆(𝑓,𝑛,𝑝)   𝑁(𝑓,𝑝)

Proof of Theorem gneispacess2
StepHypRef Expression
1 gneispace.a . . . . 5 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}
21gneispacess 40767 . . . 4 (𝐹𝐴 → ∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)))
3 fveq2 6661 . . . . . 6 (𝑝 = 𝑃 → (𝐹𝑝) = (𝐹𝑃))
43eleq2d 2901 . . . . . . . 8 (𝑝 = 𝑃 → (𝑠 ∈ (𝐹𝑝) ↔ 𝑠 ∈ (𝐹𝑃)))
54imbi2d 344 . . . . . . 7 (𝑝 = 𝑃 → ((𝑛𝑠𝑠 ∈ (𝐹𝑝)) ↔ (𝑛𝑠𝑠 ∈ (𝐹𝑃))))
65ralbidv 3192 . . . . . 6 (𝑝 = 𝑃 → (∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)) ↔ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑃))))
73, 6raleqbidv 3392 . . . . 5 (𝑝 = 𝑃 → (∀𝑛 ∈ (𝐹𝑝)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)) ↔ ∀𝑛 ∈ (𝐹𝑃)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑃))))
87rspccv 3606 . . . 4 (∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)) → (𝑃 ∈ dom 𝐹 → ∀𝑛 ∈ (𝐹𝑃)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑃))))
92, 8syl 17 . . 3 (𝐹𝐴 → (𝑃 ∈ dom 𝐹 → ∀𝑛 ∈ (𝐹𝑃)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑃))))
10 sseq1 3978 . . . . . . . 8 (𝑛 = 𝑁 → (𝑛𝑠𝑁𝑠))
1110imbi1d 345 . . . . . . 7 (𝑛 = 𝑁 → ((𝑛𝑠𝑠 ∈ (𝐹𝑃)) ↔ (𝑁𝑠𝑠 ∈ (𝐹𝑃))))
1211ralbidv 3192 . . . . . 6 (𝑛 = 𝑁 → (∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑃)) ↔ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑁𝑠𝑠 ∈ (𝐹𝑃))))
1312rspccv 3606 . . . . 5 (∀𝑛 ∈ (𝐹𝑃)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑃)) → (𝑁 ∈ (𝐹𝑃) → ∀𝑠 ∈ 𝒫 dom 𝐹(𝑁𝑠𝑠 ∈ (𝐹𝑃))))
14 sseq2 3979 . . . . . . 7 (𝑠 = 𝑆 → (𝑁𝑠𝑁𝑆))
15 eleq1 2903 . . . . . . 7 (𝑠 = 𝑆 → (𝑠 ∈ (𝐹𝑃) ↔ 𝑆 ∈ (𝐹𝑃)))
1614, 15imbi12d 348 . . . . . 6 (𝑠 = 𝑆 → ((𝑁𝑠𝑠 ∈ (𝐹𝑃)) ↔ (𝑁𝑆𝑆 ∈ (𝐹𝑃))))
1716rspccv 3606 . . . . 5 (∀𝑠 ∈ 𝒫 dom 𝐹(𝑁𝑠𝑠 ∈ (𝐹𝑃)) → (𝑆 ∈ 𝒫 dom 𝐹 → (𝑁𝑆𝑆 ∈ (𝐹𝑃))))
1813, 17syl6 35 . . . 4 (∀𝑛 ∈ (𝐹𝑃)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑃)) → (𝑁 ∈ (𝐹𝑃) → (𝑆 ∈ 𝒫 dom 𝐹 → (𝑁𝑆𝑆 ∈ (𝐹𝑃)))))
19183impd 1345 . . 3 (∀𝑛 ∈ (𝐹𝑃)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑃)) → ((𝑁 ∈ (𝐹𝑃) ∧ 𝑆 ∈ 𝒫 dom 𝐹𝑁𝑆) → 𝑆 ∈ (𝐹𝑃)))
209, 19syl6 35 . 2 (𝐹𝐴 → (𝑃 ∈ dom 𝐹 → ((𝑁 ∈ (𝐹𝑃) ∧ 𝑆 ∈ 𝒫 dom 𝐹𝑁𝑆) → 𝑆 ∈ (𝐹𝑃))))
2120imp31 421 1 (((𝐹𝐴𝑃 ∈ dom 𝐹) ∧ (𝑁 ∈ (𝐹𝑃) ∧ 𝑆 ∈ 𝒫 dom 𝐹𝑁𝑆)) → 𝑆 ∈ (𝐹𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  {cab 2802  wral 3133  cdif 3916  wss 3919  c0 4276  𝒫 cpw 4522  {csn 4550  dom cdm 5542  wf 6339  cfv 6343
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
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-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351
This theorem is referenced by: (None)
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