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Theorem gneispace2 38932
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 5479 . . . 4 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
32pweqd 4307 . . . . . . 7 (𝑓 = 𝐹 → 𝒫 dom 𝑓 = 𝒫 dom 𝐹)
43difeq1d 3870 . . . . . 6 (𝑓 = 𝐹 → (𝒫 dom 𝑓 ∖ {∅}) = (𝒫 dom 𝐹 ∖ {∅}))
54pweqd 4307 . . . . 5 (𝑓 = 𝐹 → 𝒫 (𝒫 dom 𝑓 ∖ {∅}) = 𝒫 (𝒫 dom 𝐹 ∖ {∅}))
65difeq1d 3870 . . . 4 (𝑓 = 𝐹 → (𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) = (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))
71, 2, 6feq123d 6195 . . 3 (𝑓 = 𝐹 → (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ↔ 𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})))
8 fveq1 6351 . . . . 5 (𝑓 = 𝐹 → (𝑓𝑝) = (𝐹𝑝))
98eleq2d 2825 . . . . . . . 8 (𝑓 = 𝐹 → (𝑠 ∈ (𝑓𝑝) ↔ 𝑠 ∈ (𝐹𝑝)))
109imbi2d 329 . . . . . . 7 (𝑓 = 𝐹 → ((𝑛𝑠𝑠 ∈ (𝑓𝑝)) ↔ (𝑛𝑠𝑠 ∈ (𝐹𝑝))))
113, 10raleqbidv 3291 . . . . . 6 (𝑓 = 𝐹 → (∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝)) ↔ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝))))
1211anbi2d 742 . . . . 5 (𝑓 = 𝐹 → ((𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))) ↔ (𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)))))
138, 12raleqbidv 3291 . . . 4 (𝑓 = 𝐹 → (∀𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))) ↔ ∀𝑛 ∈ (𝐹𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛𝑠𝑠 ∈ (𝐹𝑝)))))
142, 13raleqbidv 3291 . . 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 749 . 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 3493 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 196  wa 383   = wceq 1632  wcel 2139  {cab 2746  wral 3050  cdif 3712  wss 3715  c0 4058  𝒫 cpw 4302  {csn 4321  dom cdm 5266  wf 6045  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057
This theorem is referenced by:  gneispace3  38933  gneispacef  38935  gneispaceel  38943  gneispacess  38945
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