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Theorem gneispacess2 38963
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 38962 . . . 4 (𝐹𝐴 → ∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)))
3 fveq2 6332 . . . . . 6 (𝑝 = 𝑃 → (𝐹𝑝) = (𝐹𝑃))
43eleq2d 2835 . . . . . . . 8 (𝑝 = 𝑃 → (𝑠 ∈ (𝐹𝑝) ↔ 𝑠 ∈ (𝐹𝑃)))
54imbi2d 329 . . . . . . 7 (𝑝 = 𝑃 → ((𝑛𝑠𝑠 ∈ (𝐹𝑝)) ↔ (𝑛𝑠𝑠 ∈ (𝐹𝑃))))
65ralbidv 3134 . . . . . 6 (𝑝 = 𝑃 → (∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)) ↔ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑃))))
73, 6raleqbidv 3300 . . . . 5 (𝑝 = 𝑃 → (∀𝑛 ∈ (𝐹𝑝)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)) ↔ ∀𝑛 ∈ (𝐹𝑃)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑃))))
87rspccv 3455 . . . 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 3773 . . . . . . . 8 (𝑛 = 𝑁 → (𝑛𝑠𝑁𝑠))
1110imbi1d 330 . . . . . . 7 (𝑛 = 𝑁 → ((𝑛𝑠𝑠 ∈ (𝐹𝑃)) ↔ (𝑁𝑠𝑠 ∈ (𝐹𝑃))))
1211ralbidv 3134 . . . . . 6 (𝑛 = 𝑁 → (∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑃)) ↔ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑁𝑠𝑠 ∈ (𝐹𝑃))))
1312rspccv 3455 . . . . 5 (∀𝑛 ∈ (𝐹𝑃)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑃)) → (𝑁 ∈ (𝐹𝑃) → ∀𝑠 ∈ 𝒫 dom 𝐹(𝑁𝑠𝑠 ∈ (𝐹𝑃))))
14 sseq2 3774 . . . . . . 7 (𝑠 = 𝑆 → (𝑁𝑠𝑁𝑆))
15 eleq1 2837 . . . . . . 7 (𝑠 = 𝑆 → (𝑠 ∈ (𝐹𝑃) ↔ 𝑆 ∈ (𝐹𝑃)))
1614, 15imbi12d 333 . . . . . 6 (𝑠 = 𝑆 → ((𝑁𝑠𝑠 ∈ (𝐹𝑃)) ↔ (𝑁𝑆𝑆 ∈ (𝐹𝑃))))
1716rspccv 3455 . . . . 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 1440 . . 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 404 1 (((𝐹𝐴𝑃 ∈ dom 𝐹) ∧ (𝑁 ∈ (𝐹𝑃) ∧ 𝑆 ∈ 𝒫 dom 𝐹𝑁𝑆)) → 𝑆 ∈ (𝐹𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1070   = wceq 1630  wcel 2144  {cab 2756  wral 3060  cdif 3718  wss 3721  c0 4061  𝒫 cpw 4295  {csn 4314  dom cdm 5249  wf 6027  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039
This theorem is referenced by: (None)
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