Theorem pptbas 23381

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 23380	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)} ∈ (TopOn‘𝐴))
2		topontop 23286	. . . 4 ⊢ ({𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)} ∈ (TopOn‘𝐴) → {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)} ∈ Top)
3	1, 2	syl 18	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)} ∈ Top)
4		eleq2 2853	. . . . . . 7 ⊢ (𝑦 = {𝑥, 𝑃} → (𝑃 ∈ 𝑦 ↔ 𝑃 ∈ {𝑥, 𝑃}))
5		eqeq1 2767	. . . . . . 7 ⊢ (𝑦 = {𝑥, 𝑃} → (𝑦 = ∅ ↔ {𝑥, 𝑃} = ∅))
6	4, 5	orbi12d 932	. . . . . 6 ⊢ (𝑦 = {𝑥, 𝑃} → ((𝑃 ∈ 𝑦 ∨ 𝑦 = ∅) ↔ (𝑃 ∈ {𝑥, 𝑃} ∨ {𝑥, 𝑃} = ∅)))
7		simpr 490	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
8		simplr 781	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑃 ∈ 𝐴)
9	7, 8	prssd 4783	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → {𝑥, 𝑃} ⊆ 𝐴)
10		prex 5403	. . . . . . . 8 ⊢ {𝑥, 𝑃} ∈ V
11	10	elpw 4564	. . . . . . 7 ⊢ ({𝑥, 𝑃} ∈ 𝒫 𝐴 ↔ {𝑥, 𝑃} ⊆ 𝐴)
12	9, 11	sylibr 237	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → {𝑥, 𝑃} ∈ 𝒫 𝐴)
13		prid2g 4724	. . . . . . . 8 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ {𝑥, 𝑃})
14	13	ad2antlr 740	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑃 ∈ {𝑥, 𝑃})
15	14	orcd 887	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑃 ∈ {𝑥, 𝑃} ∨ {𝑥, 𝑃} = ∅))
16	6, 12, 15	elrabd 3650	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → {𝑥, 𝑃} ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)})
17	16	fmpttd 7147	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃}):𝐴⟶{𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)})
18	17	frnd 6748	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ran (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃}) ⊆ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)})
19		eleq2 2853	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑃 ∈ 𝑦 ↔ 𝑃 ∈ 𝑧))
20		eqeq1 2767	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 = ∅ ↔ 𝑧 = ∅))
21	19, 20	orbi12d 932	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((𝑃 ∈ 𝑦 ∨ 𝑦 = ∅) ↔ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))
22	21	elrab 3648	. . . . 5 ⊢ (𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))
23		elpwi 4567	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝒫 𝐴 → 𝑧 ⊆ 𝐴)
24	23	ad2antrl 741	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) → 𝑧 ⊆ 𝐴)
25	24	sselda 3913	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → 𝑤 ∈ 𝐴)
26		prid1g 4723	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑧 → 𝑤 ∈ {𝑤, 𝑃})
27	26	adantl 487	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → 𝑤 ∈ {𝑤, 𝑃})
28		simpr 490	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → 𝑤 ∈ 𝑧)
29		n0i 4289	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝑧 → ¬ 𝑧 = ∅)
30	29	adantl 487	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → ¬ 𝑧 = ∅)
31		simplrr 790	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))
32	31	ord 878	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → (¬ 𝑃 ∈ 𝑧 → 𝑧 = ∅))
33	30, 32	mt3d 149	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → 𝑃 ∈ 𝑧)
34	28, 33	prssd 4783	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → {𝑤, 𝑃} ⊆ 𝑧)
35		preq1 4696	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → {𝑥, 𝑃} = {𝑤, 𝑃})
36	35	eleq2d 2850	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝑤 ∈ {𝑥, 𝑃} ↔ 𝑤 ∈ {𝑤, 𝑃}))
37	35	sseq1d 3944	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → ({𝑥, 𝑃} ⊆ 𝑧 ↔ {𝑤, 𝑃} ⊆ 𝑧))
38	36, 37	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → ((𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧) ↔ (𝑤 ∈ {𝑤, 𝑃} ∧ {𝑤, 𝑃} ⊆ 𝑧)))
39	38	rspcev 3580	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝐴 ∧ (𝑤 ∈ {𝑤, 𝑃} ∧ {𝑤, 𝑃} ⊆ 𝑧)) → ∃𝑥 ∈ 𝐴 (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧))
40	25, 27, 34, 39	syl12anc 850	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → ∃𝑥 ∈ 𝐴 (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧))
41	10	rgenw 3084	. . . . . . . . 9 ⊢ ∀𝑥 ∈ 𝐴 {𝑥, 𝑃} ∈ V
42		eqid 2763	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃}) = (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃})
43		eleq2 2853	. . . . . . . . . . 11 ⊢ (𝑣 = {𝑥, 𝑃} → (𝑤 ∈ 𝑣 ↔ 𝑤 ∈ {𝑥, 𝑃}))
44		sseq1 3938	. . . . . . . . . . 11 ⊢ (𝑣 = {𝑥, 𝑃} → (𝑣 ⊆ 𝑧 ↔ {𝑥, 𝑃} ⊆ 𝑧))
45	43, 44	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑣 = {𝑥, 𝑃} → ((𝑤 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑧) ↔ (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧)))
46	42, 45	rexrnmptw 7127	. . . . . . . . 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 237	. . . . . . 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 3158	. . . . . 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 418	. . . . 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 245	. . . 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 3157	. . 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 23362	. . 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 1399	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → (topGen‘ran (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃})) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)})
55		eleq2 2853	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑃 ∈ 𝑦 ↔ 𝑃 ∈ 𝑥))
56		eqeq1 2767	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = ∅ ↔ 𝑥 = ∅))
57	55, 56	orbi12d 932	. . 3 ⊢ (𝑦 = 𝑥 → ((𝑃 ∈ 𝑦 ∨ 𝑦 = ∅) ↔ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)))
58	57	cbvrabv 3426	. 2 ⊢ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)} = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}
59	54, 58	eqtr2di 2815	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} = (topGen‘ran (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃})))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  𝒫 cpw 4560  {cpr 4590   ↦ cmpt 5192  ran crn 5659  ‘cfv 6566  topGenctg 17658  Topctop 23266  TopOnctopon 23283

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-topgen 17664  df-top 23267  df-topon 23284

This theorem is used by: (None)