Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  gneispacess2 Structured version   Visualization version   GIF version

Theorem gneispacess2 39050
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 39049 . . . 4 (𝐹𝐴 → ∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)))
3 fveq2 6375 . . . . . 6 (𝑝 = 𝑃 → (𝐹𝑝) = (𝐹𝑃))
43eleq2d 2830 . . . . . . . 8 (𝑝 = 𝑃 → (𝑠 ∈ (𝐹𝑝) ↔ 𝑠 ∈ (𝐹𝑃)))
54imbi2d 331 . . . . . . 7 (𝑝 = 𝑃 → ((𝑛𝑠𝑠 ∈ (𝐹𝑝)) ↔ (𝑛𝑠𝑠 ∈ (𝐹𝑃))))
65ralbidv 3133 . . . . . 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 3458 . . . 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 3786 . . . . . . . 8 (𝑛 = 𝑁 → (𝑛𝑠𝑁𝑠))
1110imbi1d 332 . . . . . . 7 (𝑛 = 𝑁 → ((𝑛𝑠𝑠 ∈ (𝐹𝑃)) ↔ (𝑁𝑠𝑠 ∈ (𝐹𝑃))))
1211ralbidv 3133 . . . . . 6 (𝑛 = 𝑁 → (∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑃)) ↔ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑁𝑠𝑠 ∈ (𝐹𝑃))))
1312rspccv 3458 . . . . 5 (∀𝑛 ∈ (𝐹𝑃)∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑃)) → (𝑁 ∈ (𝐹𝑃) → ∀𝑠 ∈ 𝒫 dom 𝐹(𝑁𝑠𝑠 ∈ (𝐹𝑃))))
14 sseq2 3787 . . . . . . 7 (𝑠 = 𝑆 → (𝑁𝑠𝑁𝑆))
15 eleq1 2832 . . . . . . 7 (𝑠 = 𝑆 → (𝑠 ∈ (𝐹𝑃) ↔ 𝑆 ∈ (𝐹𝑃)))
1614, 15imbi12d 335 . . . . . 6 (𝑠 = 𝑆 → ((𝑁𝑠𝑠 ∈ (𝐹𝑃)) ↔ (𝑁𝑆𝑆 ∈ (𝐹𝑃))))
1716rspccv 3458 . . . . 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 1457 . . 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 408 1 (((𝐹𝐴𝑃 ∈ dom 𝐹) ∧ (𝑁 ∈ (𝐹𝑃) ∧ 𝑆 ∈ 𝒫 dom 𝐹𝑁𝑆)) → 𝑆 ∈ (𝐹𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107   = wceq 1652  wcel 2155  {cab 2751  wral 3055  cdif 3729  wss 3732  c0 4079  𝒫 cpw 4315  {csn 4334  dom cdm 5277  wf 6064  cfv 6068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fv 6076
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator