Theorem pptbas 23031

Description: The particular point topology is generated by a basis consisting of pairs {𝑥, 𝑃} for each 𝑥 ∈ 𝐴. (Contributed by Mario Carneiro, 3-Sep-2015.)

Assertion

Ref	Expression
pptbas	⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} = (topGen‘ran (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃})))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑃   𝑥,𝑉

Proof of Theorem pptbas

Dummy variables 𝑤 𝑣 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ppttop 23030	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)} ∈ (TopOn‘𝐴))
2		topontop 22935	. . . 4 ⊢ ({𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)} ∈ (TopOn‘𝐴) → {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)} ∈ Top)
3	1, 2	syl 17	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)} ∈ Top)
4		eleq2 2828	. . . . . . 7 ⊢ (𝑦 = {𝑥, 𝑃} → (𝑃 ∈ 𝑦 ↔ 𝑃 ∈ {𝑥, 𝑃}))
5		eqeq1 2739	. . . . . . 7 ⊢ (𝑦 = {𝑥, 𝑃} → (𝑦 = ∅ ↔ {𝑥, 𝑃} = ∅))
6	4, 5	orbi12d 918	. . . . . 6 ⊢ (𝑦 = {𝑥, 𝑃} → ((𝑃 ∈ 𝑦 ∨ 𝑦 = ∅) ↔ (𝑃 ∈ {𝑥, 𝑃} ∨ {𝑥, 𝑃} = ∅)))
7		simpr 484	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
8		simplr 769	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑃 ∈ 𝐴)
9	7, 8	prssd 4827	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → {𝑥, 𝑃} ⊆ 𝐴)
10		prex 5443	. . . . . . . 8 ⊢ {𝑥, 𝑃} ∈ V
11	10	elpw 4609	. . . . . . 7 ⊢ ({𝑥, 𝑃} ∈ 𝒫 𝐴 ↔ {𝑥, 𝑃} ⊆ 𝐴)
12	9, 11	sylibr 234	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → {𝑥, 𝑃} ∈ 𝒫 𝐴)
13		prid2g 4766	. . . . . . . 8 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ {𝑥, 𝑃})
14	13	ad2antlr 727	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑃 ∈ {𝑥, 𝑃})
15	14	orcd 873	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑃 ∈ {𝑥, 𝑃} ∨ {𝑥, 𝑃} = ∅))
16	6, 12, 15	elrabd 3697	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → {𝑥, 𝑃} ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)})
17	16	fmpttd 7135	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃}):𝐴⟶{𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)})
18	17	frnd 6745	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ran (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃}) ⊆ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)})
19		eleq2 2828	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑃 ∈ 𝑦 ↔ 𝑃 ∈ 𝑧))
20		eqeq1 2739	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 = ∅ ↔ 𝑧 = ∅))
21	19, 20	orbi12d 918	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((𝑃 ∈ 𝑦 ∨ 𝑦 = ∅) ↔ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))
22	21	elrab 3695	. . . . 5 ⊢ (𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))
23		elpwi 4612	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝒫 𝐴 → 𝑧 ⊆ 𝐴)
24	23	ad2antrl 728	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) → 𝑧 ⊆ 𝐴)
25	24	sselda 3995	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → 𝑤 ∈ 𝐴)
26		prid1g 4765	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑧 → 𝑤 ∈ {𝑤, 𝑃})
27	26	adantl 481	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → 𝑤 ∈ {𝑤, 𝑃})
28		simpr 484	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → 𝑤 ∈ 𝑧)
29		n0i 4346	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝑧 → ¬ 𝑧 = ∅)
30	29	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → ¬ 𝑧 = ∅)
31		simplrr 778	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))
32	31	ord 864	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → (¬ 𝑃 ∈ 𝑧 → 𝑧 = ∅))
33	30, 32	mt3d 148	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → 𝑃 ∈ 𝑧)
34	28, 33	prssd 4827	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → {𝑤, 𝑃} ⊆ 𝑧)
35		preq1 4738	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → {𝑥, 𝑃} = {𝑤, 𝑃})
36	35	eleq2d 2825	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝑤 ∈ {𝑥, 𝑃} ↔ 𝑤 ∈ {𝑤, 𝑃}))
37	35	sseq1d 4027	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ({𝑥, 𝑃} ⊆ 𝑧 ↔ {𝑤, 𝑃} ⊆ 𝑧))
38	36, 37	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → ((𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧) ↔ (𝑤 ∈ {𝑤, 𝑃} ∧ {𝑤, 𝑃} ⊆ 𝑧)))
39	38	rspcev 3622	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝐴 ∧ (𝑤 ∈ {𝑤, 𝑃} ∧ {𝑤, 𝑃} ⊆ 𝑧)) → ∃𝑥 ∈ 𝐴 (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧))
40	25, 27, 34, 39	syl12anc 837	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → ∃𝑥 ∈ 𝐴 (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧))
41	10	rgenw 3063	. . . . . . . . 9 ⊢ ∀𝑥 ∈ 𝐴 {𝑥, 𝑃} ∈ V
42		eqid 2735	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃}) = (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃})
43		eleq2 2828	. . . . . . . . . . 11 ⊢ (𝑣 = {𝑥, 𝑃} → (𝑤 ∈ 𝑣 ↔ 𝑤 ∈ {𝑥, 𝑃}))
44		sseq1 4021	. . . . . . . . . . 11 ⊢ (𝑣 = {𝑥, 𝑃} → (𝑣 ⊆ 𝑧 ↔ {𝑥, 𝑃} ⊆ 𝑧))
45	43, 44	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑣 = {𝑥, 𝑃} → ((𝑤 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧) ↔ (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧)))
46	42, 45	rexrnmptw 7115	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 {𝑥, 𝑃} ∈ V → (∃𝑣 ∈ ran (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃})(𝑤 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧) ↔ ∃𝑥 ∈ 𝐴 (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧)))
47	41, 46	ax-mp 5	. . . . . . . 8 ⊢ (∃𝑣 ∈ ran (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃})(𝑤 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧) ↔ ∃𝑥 ∈ 𝐴 (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧))
48	40, 47	sylibr 234	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → ∃𝑣 ∈ ran (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃})(𝑤 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧))
49	48	ralrimiva 3144	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) → ∀𝑤 ∈ 𝑧 ∃𝑣 ∈ ran (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃})(𝑤 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧))
50	49	ex 412	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ((𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)) → ∀𝑤 ∈ 𝑧 ∃𝑣 ∈ ran (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃})(𝑤 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧)))
51	22, 50	biimtrid 242	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → (𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)} → ∀𝑤 ∈ 𝑧 ∃𝑣 ∈ ran (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃})(𝑤 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧)))
52	51	ralrimiv 3143	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ∀𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)}∀𝑤 ∈ 𝑧 ∃𝑣 ∈ ran (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃})(𝑤 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧))
53		basgen2 23012	. . 3 ⊢ (({𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)} ∈ Top ∧ ran (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃}) ⊆ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)} ∧ ∀𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)}∀𝑤 ∈ 𝑧 ∃𝑣 ∈ ran (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃})(𝑤 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧)) → (topGen‘ran (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃})) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)})
54	3, 18, 52, 53	syl3anc 1370	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → (topGen‘ran (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃})) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)})
55		eleq2 2828	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑃 ∈ 𝑦 ↔ 𝑃 ∈ 𝑥))
56		eqeq1 2739	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = ∅ ↔ 𝑥 = ∅))
57	55, 56	orbi12d 918	. . 3 ⊢ (𝑦 = 𝑥 → ((𝑃 ∈ 𝑦 ∨ 𝑦 = ∅) ↔ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)))
58	57	cbvrabv 3444	. 2 ⊢ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)} = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}
59	54, 58	eqtr2di 2792	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} = (topGen‘ran (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  {crab 3433  Vcvv 3478   ⊆ wss 3963  ∅c0 4339  𝒫 cpw 4605  {cpr 4633   ↦ cmpt 5231  ran crn 5690  ‘cfv 6563  topGenctg 17484  Topctop 22915  TopOnctopon 22932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-topgen 17490  df-top 22916  df-topon 22933

This theorem is referenced by: (None)