Theorem opnfbas 23807

Description: The collection of open supersets of a nonempty set in a topology is a neighborhoods of the set, one of the motivations for the filter concept. (Contributed by Jeff Hankins, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Aug-2015.)

Hypothesis

Ref	Expression
opnfbas.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
opnfbas	⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∈ (fBas‘𝑋))

Distinct variable groups:   𝑥,𝐽   𝑥,𝑆   𝑥,𝑋

Proof of Theorem opnfbas

Dummy variables 𝑠 𝑟 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 4020	. . . 4 ⊢ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ⊆ 𝐽
2		opnfbas.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	2	eqimss2i 3983	. . . . 5 ⊢ ∪ 𝐽 ⊆ 𝑋
4		sspwuni 5042	. . . . 5 ⊢ (𝐽 ⊆ 𝒫 𝑋 ↔ ∪ 𝐽 ⊆ 𝑋)
5	3, 4	mpbir 231	. . . 4 ⊢ 𝐽 ⊆ 𝒫 𝑋
6	1, 5	sstri 3931	. . 3 ⊢ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ⊆ 𝒫 𝑋
7	6	a1i 11	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ⊆ 𝒫 𝑋)
8	2	topopn 22871	. . . . . . 7 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
9	8	anim1i 616	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑋))
10	9	3adant3 1133	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (𝑋 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑋))
11		sseq2 3948	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑆 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑋))
12	11	elrab 3634	. . . . 5 ⊢ (𝑋 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ↔ (𝑋 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑋))
13	10, 12	sylibr 234	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → 𝑋 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥})
14	13	ne0d 4282	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ≠ ∅)
15		ss0 4342	. . . . . . 7 ⊢ (𝑆 ⊆ ∅ → 𝑆 = ∅)
16	15	necon3ai 2957	. . . . . 6 ⊢ (𝑆 ≠ ∅ → ¬ 𝑆 ⊆ ∅)
17	16	3ad2ant3 1136	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ¬ 𝑆 ⊆ ∅)
18	17	intnand 488	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ¬ (∅ ∈ 𝐽 ∧ 𝑆 ⊆ ∅))
19		df-nel 3037	. . . . 5 ⊢ (∅ ∉ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ↔ ¬ ∅ ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥})
20		sseq2 3948	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑆 ⊆ 𝑥 ↔ 𝑆 ⊆ ∅))
21	20	elrab 3634	. . . . . 6 ⊢ (∅ ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ↔ (∅ ∈ 𝐽 ∧ 𝑆 ⊆ ∅))
22	21	notbii 320	. . . . 5 ⊢ (¬ ∅ ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ↔ ¬ (∅ ∈ 𝐽 ∧ 𝑆 ⊆ ∅))
23	19, 22	bitr2i 276	. . . 4 ⊢ (¬ (∅ ∈ 𝐽 ∧ 𝑆 ⊆ ∅) ↔ ∅ ∉ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥})
24	18, 23	sylib 218	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ∅ ∉ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥})
25		sseq2 3948	. . . . . . 7 ⊢ (𝑥 = 𝑟 → (𝑆 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑟))
26	25	elrab 3634	. . . . . 6 ⊢ (𝑟 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ↔ (𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟))
27		sseq2 3948	. . . . . . 7 ⊢ (𝑥 = 𝑠 → (𝑆 ⊆ 𝑥 ↔ 𝑆 ⊆ 𝑠))
28	27	elrab 3634	. . . . . 6 ⊢ (𝑠 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ↔ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠))
29	26, 28	anbi12i 629	. . . . 5 ⊢ ((𝑟 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∧ 𝑠 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}) ↔ ((𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟) ∧ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠)))
30		simpl 482	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ ((𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟) ∧ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠))) → 𝐽 ∈ Top)
31		simprll 779	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ ((𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟) ∧ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠))) → 𝑟 ∈ 𝐽)
32		simprrl 781	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ ((𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟) ∧ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠))) → 𝑠 ∈ 𝐽)
33		inopn 22864	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑟 ∈ 𝐽 ∧ 𝑠 ∈ 𝐽) → (𝑟 ∩ 𝑠) ∈ 𝐽)
34	30, 31, 32, 33	syl3anc 1374	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ ((𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟) ∧ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠))) → (𝑟 ∩ 𝑠) ∈ 𝐽)
35		ssin 4179	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ 𝑟 ∧ 𝑆 ⊆ 𝑠) ↔ 𝑆 ⊆ (𝑟 ∩ 𝑠))
36	35	biimpi 216	. . . . . . . . . . . 12 ⊢ ((𝑆 ⊆ 𝑟 ∧ 𝑆 ⊆ 𝑠) → 𝑆 ⊆ (𝑟 ∩ 𝑠))
37	36	ad2ant2l 747	. . . . . . . . . . 11 ⊢ (((𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟) ∧ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠)) → 𝑆 ⊆ (𝑟 ∩ 𝑠))
38	37	adantl 481	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ ((𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟) ∧ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠))) → 𝑆 ⊆ (𝑟 ∩ 𝑠))
39	34, 38	jca 511	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ ((𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟) ∧ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠))) → ((𝑟 ∩ 𝑠) ∈ 𝐽 ∧ 𝑆 ⊆ (𝑟 ∩ 𝑠)))
40	39	3ad2antl1 1187	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) ∧ ((𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟) ∧ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠))) → ((𝑟 ∩ 𝑠) ∈ 𝐽 ∧ 𝑆 ⊆ (𝑟 ∩ 𝑠)))
41		sseq2 3948	. . . . . . . . 9 ⊢ (𝑥 = (𝑟 ∩ 𝑠) → (𝑆 ⊆ 𝑥 ↔ 𝑆 ⊆ (𝑟 ∩ 𝑠)))
42	41	elrab 3634	. . . . . . . 8 ⊢ ((𝑟 ∩ 𝑠) ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ↔ ((𝑟 ∩ 𝑠) ∈ 𝐽 ∧ 𝑆 ⊆ (𝑟 ∩ 𝑠)))
43	40, 42	sylibr 234	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) ∧ ((𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟) ∧ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠))) → (𝑟 ∩ 𝑠) ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥})
44		ssid 3944	. . . . . . 7 ⊢ (𝑟 ∩ 𝑠) ⊆ (𝑟 ∩ 𝑠)
45		sseq1 3947	. . . . . . . 8 ⊢ (𝑡 = (𝑟 ∩ 𝑠) → (𝑡 ⊆ (𝑟 ∩ 𝑠) ↔ (𝑟 ∩ 𝑠) ⊆ (𝑟 ∩ 𝑠)))
46	45	rspcev 3564	. . . . . . 7 ⊢ (((𝑟 ∩ 𝑠) ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∧ (𝑟 ∩ 𝑠) ⊆ (𝑟 ∩ 𝑠)) → ∃𝑡 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}𝑡 ⊆ (𝑟 ∩ 𝑠))
47	43, 44, 46	sylancl 587	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) ∧ ((𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟) ∧ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠))) → ∃𝑡 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}𝑡 ⊆ (𝑟 ∩ 𝑠))
48	47	ex 412	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (((𝑟 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑟) ∧ (𝑠 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑠)) → ∃𝑡 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}𝑡 ⊆ (𝑟 ∩ 𝑠)))
49	29, 48	biimtrid 242	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ((𝑟 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∧ 𝑠 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}) → ∃𝑡 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}𝑡 ⊆ (𝑟 ∩ 𝑠)))
50	49	ralrimivv 3178	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ∀𝑟 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}∀𝑠 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}∃𝑡 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}𝑡 ⊆ (𝑟 ∩ 𝑠))
51	14, 24, 50	3jca 1129	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ({𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ≠ ∅ ∧ ∅ ∉ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∧ ∀𝑟 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}∀𝑠 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}∃𝑡 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}𝑡 ⊆ (𝑟 ∩ 𝑠)))
52		isfbas2 23800	. . . 4 ⊢ (𝑋 ∈ 𝐽 → ({𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∈ (fBas‘𝑋) ↔ ({𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ⊆ 𝒫 𝑋 ∧ ({𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ≠ ∅ ∧ ∅ ∉ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∧ ∀𝑟 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}∀𝑠 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}∃𝑡 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}𝑡 ⊆ (𝑟 ∩ 𝑠)))))
53	8, 52	syl 17	. . 3 ⊢ (𝐽 ∈ Top → ({𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∈ (fBas‘𝑋) ↔ ({𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ⊆ 𝒫 𝑋 ∧ ({𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ≠ ∅ ∧ ∅ ∉ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∧ ∀𝑟 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}∀𝑠 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}∃𝑡 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}𝑡 ⊆ (𝑟 ∩ 𝑠)))))
54	53	3ad2ant1 1134	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ({𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∈ (fBas‘𝑋) ↔ ({𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ⊆ 𝒫 𝑋 ∧ ({𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ≠ ∅ ∧ ∅ ∉ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∧ ∀𝑟 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}∀𝑠 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}∃𝑡 ∈ {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥}𝑡 ⊆ (𝑟 ∩ 𝑠)))))
55	7, 51, 54	mpbir2and 714	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → {𝑥 ∈ 𝐽 ∣ 𝑆 ⊆ 𝑥} ∈ (fBas‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932   ∉ wnel 3036  ∀wral 3051  ∃wrex 3061  {crab 3389   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273  𝒫 cpw 4541  ∪ cuni 4850  ‘cfv 6498  fBascfbas 21340  Topctop 22858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fv 6506  df-fbas 21349  df-top 22859

This theorem is referenced by:  neifg  36553