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Theorem gneispacess2 39285
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 39284 . . . 4 (𝐹𝐴 → ∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)))
3 fveq2 6434 . . . . . 6 (𝑝 = 𝑃 → (𝐹𝑝) = (𝐹𝑃))
43eleq2d 2893 . . . . . . . 8 (𝑝 = 𝑃 → (𝑠 ∈ (𝐹𝑝) ↔ 𝑠 ∈ (𝐹𝑃)))
54imbi2d 332 . . . . . . 7 (𝑝 = 𝑃 → ((𝑛𝑠𝑠 ∈ (𝐹𝑝)) ↔ (𝑛𝑠𝑠 ∈ (𝐹𝑃))))
65ralbidv 3196 . . . . . 6 (𝑝 = 𝑃 → (∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)) ↔ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑃))))
73, 6raleqbidv 3365 . . . . 5 (𝑝 = 𝑃 → (∀𝑛 ∈ (𝐹𝑝)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)) ↔ ∀𝑛 ∈ (𝐹𝑃)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑃))))
87rspccv 3524 . . . 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 3852 . . . . . . . 8 (𝑛 = 𝑁 → (𝑛𝑠𝑁𝑠))
1110imbi1d 333 . . . . . . 7 (𝑛 = 𝑁 → ((𝑛𝑠𝑠 ∈ (𝐹𝑃)) ↔ (𝑁𝑠𝑠 ∈ (𝐹𝑃))))
1211ralbidv 3196 . . . . . 6 (𝑛 = 𝑁 → (∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑃)) ↔ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑁𝑠𝑠 ∈ (𝐹𝑃))))
1312rspccv 3524 . . . . 5 (∀𝑛 ∈ (𝐹𝑃)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑃)) → (𝑁 ∈ (𝐹𝑃) → ∀𝑠 ∈ 𝒫 dom 𝐹(𝑁𝑠𝑠 ∈ (𝐹𝑃))))
14 sseq2 3853 . . . . . . 7 (𝑠 = 𝑆 → (𝑁𝑠𝑁𝑆))
15 eleq1 2895 . . . . . . 7 (𝑠 = 𝑆 → (𝑠 ∈ (𝐹𝑃) ↔ 𝑆 ∈ (𝐹𝑃)))
1614, 15imbi12d 336 . . . . . 6 (𝑠 = 𝑆 → ((𝑁𝑠𝑠 ∈ (𝐹𝑃)) ↔ (𝑁𝑆𝑆 ∈ (𝐹𝑃))))
1716rspccv 3524 . . . . 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 1463 . . 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 410 1 (((𝐹𝐴𝑃 ∈ dom 𝐹) ∧ (𝑁 ∈ (𝐹𝑃) ∧ 𝑆 ∈ 𝒫 dom 𝐹𝑁𝑆)) → 𝑆 ∈ (𝐹𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1113   = wceq 1658  wcel 2166  {cab 2812  wral 3118  cdif 3796  wss 3799  c0 4145  𝒫 cpw 4379  {csn 4398  dom cdm 5343  wf 6120  cfv 6124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-fv 6132
This theorem is referenced by: (None)
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