Theorem pptbas 21618

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 21617	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)} ∈ (TopOn‘𝐴))
2		topontop 21523	. . . 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 2903	. . . . . . 7 ⊢ (𝑦 = {𝑥, 𝑃} → (𝑃 ∈ 𝑦 ↔ 𝑃 ∈ {𝑥, 𝑃}))
5		eqeq1 2827	. . . . . . 7 ⊢ (𝑦 = {𝑥, 𝑃} → (𝑦 = ∅ ↔ {𝑥, 𝑃} = ∅))
6	4, 5	orbi12d 915	. . . . . 6 ⊢ (𝑦 = {𝑥, 𝑃} → ((𝑃 ∈ 𝑦 ∨ 𝑦 = ∅) ↔ (𝑃 ∈ {𝑥, 𝑃} ∨ {𝑥, 𝑃} = ∅)))
7		simpr 487	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
8		simplr 767	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑃 ∈ 𝐴)
9	7, 8	prssd 4757	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → {𝑥, 𝑃} ⊆ 𝐴)
10		prex 5335	. . . . . . . 8 ⊢ {𝑥, 𝑃} ∈ V
11	10	elpw 4545	. . . . . . 7 ⊢ ({𝑥, 𝑃} ∈ 𝒫 𝐴 ↔ {𝑥, 𝑃} ⊆ 𝐴)
12	9, 11	sylibr 236	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → {𝑥, 𝑃} ∈ 𝒫 𝐴)
13		prid2g 4699	. . . . . . . 8 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ {𝑥, 𝑃})
14	13	ad2antlr 725	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑃 ∈ {𝑥, 𝑃})
15	14	orcd 869	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑃 ∈ {𝑥, 𝑃} ∨ {𝑥, 𝑃} = ∅))
16	6, 12, 15	elrabd 3684	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → {𝑥, 𝑃} ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)})
17	16	fmpttd 6881	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃}):𝐴⟶{𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)})
18	17	frnd 6523	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ran (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃}) ⊆ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)})
19		eleq2 2903	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑃 ∈ 𝑦 ↔ 𝑃 ∈ 𝑧))
20		eqeq1 2827	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 = ∅ ↔ 𝑧 = ∅))
21	19, 20	orbi12d 915	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((𝑃 ∈ 𝑦 ∨ 𝑦 = ∅) ↔ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))
22	21	elrab 3682	. . . . 5 ⊢ (𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅)))
23		elpwi 4550	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝒫 𝐴 → 𝑧 ⊆ 𝐴)
24	23	ad2antrl 726	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) → 𝑧 ⊆ 𝐴)
25	24	sselda 3969	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → 𝑤 ∈ 𝐴)
26		prid1g 4698	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝑧 → 𝑤 ∈ {𝑤, 𝑃})
27	26	adantl 484	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → 𝑤 ∈ {𝑤, 𝑃})
28		simpr 487	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → 𝑤 ∈ 𝑧)
29		n0i 4301	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝑧 → ¬ 𝑧 = ∅)
30	29	adantl 484	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → ¬ 𝑧 = ∅)
31		simplrr 776	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))
32	31	ord 860	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → (¬ 𝑃 ∈ 𝑧 → 𝑧 = ∅))
33	30, 32	mt3d 150	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → 𝑃 ∈ 𝑧)
34	28, 33	prssd 4757	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → {𝑤, 𝑃} ⊆ 𝑧)
35		preq1 4671	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → {𝑥, 𝑃} = {𝑤, 𝑃})
36	35	eleq2d 2900	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝑤 ∈ {𝑥, 𝑃} ↔ 𝑤 ∈ {𝑤, 𝑃}))
37	35	sseq1d 4000	. . . . . . . . . . 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 3625	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝐴 ∧ (𝑤 ∈ {𝑤, 𝑃} ∧ {𝑤, 𝑃} ⊆ 𝑧)) → ∃𝑥 ∈ 𝐴 (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧))
40	25, 27, 34, 39	syl12anc 834	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 ∨ 𝑧 = ∅))) ∧ 𝑤 ∈ 𝑧) → ∃𝑥 ∈ 𝐴 (𝑤 ∈ {𝑥, 𝑃} ∧ {𝑥, 𝑃} ⊆ 𝑧))
41	10	rgenw 3152	. . . . . . . . 9 ⊢ ∀𝑥 ∈ 𝐴 {𝑥, 𝑃} ∈ V
42		eqid 2823	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃}) = (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃})
43		eleq2 2903	. . . . . . . . . . 11 ⊢ (𝑣 = {𝑥, 𝑃} → (𝑤 ∈ 𝑣 ↔ 𝑤 ∈ {𝑥, 𝑃}))
44		sseq1 3994	. . . . . . . . . . 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 6863	. . . . . . . . 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 236	. . . . . . 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 3184	. . . . . 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 415	. . . . 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	syl5bi 244	. . . 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 3183	. . 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 21599	. . 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 1367	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → (topGen‘ran (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃})) = {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)})
55		eleq2 2903	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑃 ∈ 𝑦 ↔ 𝑃 ∈ 𝑥))
56		eqeq1 2827	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 = ∅ ↔ 𝑥 = ∅))
57	55, 56	orbi12d 915	. . 3 ⊢ (𝑦 = 𝑥 → ((𝑃 ∈ 𝑦 ∨ 𝑦 = ∅) ↔ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)))
58	57	cbvrabv 3493	. 2 ⊢ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑦 ∨ 𝑦 = ∅)} = {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)}
59	54, 58	syl6req 2875	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 ∨ 𝑥 = ∅)} = (topGen‘ran (𝑥 ∈ 𝐴 ↦ {𝑥, 𝑃})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141  {crab 3144  Vcvv 3496   ⊆ wss 3938  ∅c0 4293  𝒫 cpw 4541  {cpr 4571   ↦ cmpt 5148  ran crn 5558  ‘cfv 6357  topGenctg 16713  Topctop 21503  TopOnctopon 21520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-topgen 16719  df-top 21504  df-topon 21521

This theorem is referenced by: (None)