Theorem pptbas 23036

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 23035	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)} ∈ (TopOn‘𝐴))
2		topontop 22940	. . . 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 2833	. . . . . . 7 ⊢ (𝑦 = {𝑥, 𝑃} → (𝑃 ∈ 𝑦 ↔ 𝑃 ∈ {𝑥, 𝑃}))
5		eqeq1 2744	. . . . . . 7 ⊢ (𝑦 = {𝑥, 𝑃} → (𝑦 = ∅ ↔ {𝑥, 𝑃} = ∅))
6	4, 5	orbi12d 917	. . . . . 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 768	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑃 ∈ 𝐴)
9	7, 8	prssd 4847	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → {𝑥, 𝑃} ⊆ 𝐴)
10		prex 5452	. . . . . . . 8 ⊢ {𝑥, 𝑃} ∈ V
11	10	elpw 4626	. . . . . . 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 4786	. . . . . . . 8 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ {𝑥, 𝑃})
14	13	ad2antlr 726	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑃 ∈ {𝑥, 𝑃})
15	14	orcd 872	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑃 ∈ {𝑥, 𝑃} ∨ {𝑥, 𝑃} = ∅))
16	6, 12, 15	elrabd 3710	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → {𝑥, 𝑃} ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)})
17	16	fmpttd 7149	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃}):𝐴⟶{𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)})
18	17	frnd 6755	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ran (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃}) ⊆ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)})
19		eleq2 2833	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑃 ∈ 𝑦 ↔ 𝑃 ∈ 𝑧))
20		eqeq1 2744	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 = ∅ ↔ 𝑧 = ∅))
21	19, 20	orbi12d 917	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((𝑃 ∈ 𝑦 ∨ 𝑦 = ∅) ↔ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))
22	21	elrab 3708	. . . . 5 ⊢ (𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))
23		elpwi 4629	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝒫 𝐴 → 𝑧 ⊆ 𝐴)
24	23	ad2antrl 727	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) → 𝑧 ⊆ 𝐴)
25	24	sselda 4008	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → 𝑤 ∈ 𝐴)
26		prid1g 4785	. . . . . . . . . 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 4363	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝑧 → ¬ 𝑧 = ∅)
30	29	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → ¬ 𝑧 = ∅)
31		simplrr 777	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))
32	31	ord 863	. . . . . . . . . . 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 4847	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → {𝑤, 𝑃} ⊆ 𝑧)
35		preq1 4758	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → {𝑥, 𝑃} = {𝑤, 𝑃})
36	35	eleq2d 2830	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝑤 ∈ {𝑥, 𝑃} ↔ 𝑤 ∈ {𝑤, 𝑃}))
37	35	sseq1d 4040	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ({𝑥, 𝑃} ⊆ 𝑧 ↔ {𝑤, 𝑃} ⊆ 𝑧))
38	36, 37	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → ((𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧) ↔ (𝑤 ∈ {𝑤, 𝑃} ∧ {𝑤, 𝑃} ⊆ 𝑧)))
39	38	rspcev 3635	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝐴 ∧ (𝑤 ∈ {𝑤, 𝑃} ∧ {𝑤, 𝑃} ⊆ 𝑧)) → ∃𝑥 ∈ 𝐴 (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧))
40	25, 27, 34, 39	syl12anc 836	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → ∃𝑥 ∈ 𝐴 (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧))
41	10	rgenw 3071	. . . . . . . . 9 ⊢ ∀𝑥 ∈ 𝐴 {𝑥, 𝑃} ∈ V
42		eqid 2740	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃}) = (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃})
43		eleq2 2833	. . . . . . . . . . 11 ⊢ (𝑣 = {𝑥, 𝑃} → (𝑤 ∈ 𝑣 ↔ 𝑤 ∈ {𝑥, 𝑃}))
44		sseq1 4034	. . . . . . . . . . 11 ⊢ (𝑣 = {𝑥, 𝑃} → (𝑣 ⊆ 𝑧 ↔ {𝑥, 𝑃} ⊆ 𝑧))
45	43, 44	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑣 = {𝑥, 𝑃} → ((𝑤 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧) ↔ (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧)))
46	42, 45	rexrnmptw 7129	. . . . . . . . 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 3152	. . . . . 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 3151	. . 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 23017	. . 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 1371	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → (topGen‘ran (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃})) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)})
55		eleq2 2833	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑃 ∈ 𝑦 ↔ 𝑃 ∈ 𝑥))
56		eqeq1 2744	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = ∅ ↔ 𝑥 = ∅))
57	55, 56	orbi12d 917	. . 3 ⊢ (𝑦 = 𝑥 → ((𝑃 ∈ 𝑦 ∨ 𝑦 = ∅) ↔ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)))
58	57	cbvrabv 3454	. 2 ⊢ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)} = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}
59	54, 58	eqtr2di 2797	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 846   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  {crab 3443  Vcvv 3488   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  {cpr 4650   ↦ cmpt 5249  ran crn 5701  ‘cfv 6573  topGenctg 17497  Topctop 22920  TopOnctopon 22937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-topgen 17503  df-top 22921  df-topon 22938

This theorem is referenced by: (None)