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Theorem gneispacef2 44149
Description: A generic neighborhood space is a function with a range that is a subset of the powerset of the powerset of its domain. (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
gneispacef2 (𝐹𝐴𝐹:dom 𝐹⟶𝒫 𝒫 dom 𝐹)
Distinct variable groups:   𝑛,𝐹,𝑝,𝑓   𝐹,𝑠,𝑓   𝑓,𝑛,𝑝
Allowed substitution hints:   𝐴(𝑓,𝑛,𝑠,𝑝)

Proof of Theorem gneispacef2
StepHypRef Expression
1 elex 3501 . . . 4 (𝐹𝐴𝐹 ∈ V)
2 gneispace.a . . . . 5 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}
32gneispace 44147 . . . 4 (𝐹 ∈ V → (𝐹𝐴 ↔ (Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)))))))
41, 3syl 17 . . 3 (𝐹𝐴 → (𝐹𝐴 ↔ (Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)))))))
54ibi 267 . 2 (𝐹𝐴 → (Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝))))))
6 simp1 1137 . . . 4 ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝))))) → Fun 𝐹)
76funfnd 6597 . . 3 ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝))))) → 𝐹 Fn dom 𝐹)
8 simp2 1138 . . 3 ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝))))) → ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹)
9 df-f 6565 . . 3 (𝐹:dom 𝐹⟶𝒫 𝒫 dom 𝐹 ↔ (𝐹 Fn dom 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹))
107, 8, 9sylanbrc 583 . 2 ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝒫 𝒫 dom 𝐹 ∧ ∀𝑝 ∈ dom 𝐹((𝐹𝑝) ≠ ∅ ∧ ∀𝑛 ∈ (𝐹𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝))))) → 𝐹:dom 𝐹⟶𝒫 𝒫 dom 𝐹)
115, 10syl 17 1 (𝐹𝐴𝐹:dom 𝐹⟶𝒫 𝒫 dom 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  {cab 2714  wne 2940  wral 3061  Vcvv 3480  cdif 3948  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626  dom cdm 5685  ran crn 5686  Fun wfun 6555   Fn wfn 6556  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-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
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-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  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-mpt 5226  df-id 5578  df-xp 5691  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: (None)
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