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Theorem gneispace2 43565
Description: The predicate that 𝐹 is a (generic) Seifert and Threlfall neighborhood space. (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
gneispace2 (𝐹𝑉 → (𝐹𝐴 ↔ (𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝))))))
Distinct variable groups:   𝑛,𝐹,𝑝,𝑓   𝐹,𝑠,𝑓   𝑓,𝑛,𝑝
Allowed substitution hints:   𝐴(𝑓,𝑛,𝑠,𝑝)   𝑉(𝑓,𝑛,𝑠,𝑝)

Proof of Theorem gneispace2
StepHypRef Expression
1 id 22 . . . 4 (𝑓 = 𝐹𝑓 = 𝐹)
2 dmeq 5908 . . . 4 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
32pweqd 4621 . . . . . . 7 (𝑓 = 𝐹 → 𝒫 dom 𝑓 = 𝒫 dom 𝐹)
43difeq1d 4119 . . . . . 6 (𝑓 = 𝐹 → (𝒫 dom 𝑓 ∖ {∅}) = (𝒫 dom 𝐹 ∖ {∅}))
54pweqd 4621 . . . . 5 (𝑓 = 𝐹 → 𝒫 (𝒫 dom 𝑓 ∖ {∅}) = 𝒫 (𝒫 dom 𝐹 ∖ {∅}))
65difeq1d 4119 . . . 4 (𝑓 = 𝐹 → (𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) = (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))
71, 2, 6feq123d 6714 . . 3 (𝑓 = 𝐹 → (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ↔ 𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})))
8 fveq1 6899 . . . . 5 (𝑓 = 𝐹 → (𝑓𝑝) = (𝐹𝑝))
98eleq2d 2814 . . . . . . . 8 (𝑓 = 𝐹 → (𝑠 ∈ (𝑓𝑝) ↔ 𝑠 ∈ (𝐹𝑝)))
109imbi2d 339 . . . . . . 7 (𝑓 = 𝐹 → ((𝑛𝑠𝑠 ∈ (𝑓𝑝)) ↔ (𝑛𝑠𝑠 ∈ (𝐹𝑝))))
113, 10raleqbidv 3338 . . . . . 6 (𝑓 = 𝐹 → (∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝)) ↔ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝))))
1211anbi2d 628 . . . . 5 (𝑓 = 𝐹 → ((𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))) ↔ (𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)))))
138, 12raleqbidv 3338 . . . 4 (𝑓 = 𝐹 → (∀𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))) ↔ ∀𝑛 ∈ (𝐹𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)))))
142, 13raleqbidv 3338 . . 3 (𝑓 = 𝐹 → (∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))) ↔ ∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)))))
157, 14anbi12d 630 . 2 (𝑓 = 𝐹 → ((𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝)))) ↔ (𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝))))))
16 gneispace.a . 2 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}
1715, 16elab2g 3669 1 (𝐹𝑉 → (𝐹𝐴 ↔ (𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝐹𝑛 ∈ (𝐹𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  {cab 2704  wral 3057  cdif 3944  wss 3947  c0 4324  𝒫 cpw 4604  {csn 4630  dom cdm 5680  wf 6547  cfv 6551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3058  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fv 6559
This theorem is referenced by:  gneispace3  43566  gneispacef  43568  gneispaceel  43576  gneispacess  43578
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