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 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}

Assertion

Ref	Expression
gneispace2	⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ 𝐴 ↔ (𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))))

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

Step	Hyp	Ref	Expression
1		id 22	. . . 4 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹)
2		dmeq 5908	. . . 4 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
3	2	pweqd 4621	. . . . . . 7 ⊢ (𝑓 = 𝐹 → 𝒫 dom 𝑓 = 𝒫 dom 𝐹)
4	3	difeq1d 4119	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝒫 dom 𝑓 ∖ {∅}) = (𝒫 dom 𝐹 ∖ {∅}))
5	4	pweqd 4621	. . . . 5 ⊢ (𝑓 = 𝐹 → 𝒫 (𝒫 dom 𝑓 ∖ {∅}) = 𝒫 (𝒫 dom 𝐹 ∖ {∅}))
6	5	difeq1d 4119	. . . 4 ⊢ (𝑓 = 𝐹 → (𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) = (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))
7	1, 2, 6	feq123d 6714	. . 3 ⊢ (𝑓 = 𝐹 → (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ↔ 𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})))
8		fveq1 6899	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑝) = (𝐹‘𝑝))
9	8	eleq2d 2814	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑠 ∈ (𝑓‘𝑝) ↔ 𝑠 ∈ (𝐹‘𝑝)))
10	9	imbi2d 339	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝)) ↔ (𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))
11	3, 10	raleqbidv 3338	. . . . . 6 ⊢ (𝑓 = 𝐹 → (∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝)) ↔ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))
12	11	anbi2d 628	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))) ↔ (𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))))
13	8, 12	raleqbidv 3338	. . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))) ↔ ∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))))
14	2, 13	raleqbidv 3338	. . 3 ⊢ (𝑓 = 𝐹 → (∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))) ↔ ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝)))))
15	7, 14	anbi12d 630	. 2 ⊢ (𝑓 = 𝐹 → ((𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝)))) ↔ (𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))))
16		gneispace.a	. 2 ⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}
17	15, 16	elab2g 3669	1 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ∈ 𝐴 ↔ (𝐹:dom 𝐹⟶(𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝐹∀𝑛 ∈ (𝐹‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝐹(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝐹‘𝑝))))))

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