Theorem neips 23411

Description: A neighborhood of a set is a neighborhood of every point in the set. Proposition 1 of [BourbakiTop1] p. I.2. (Contributed by FL, 16-Nov-2006.)

Hypothesis

Ref	Expression
neips.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
neips	⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ ∀𝑝 ∈ 𝑆 𝑁 ∈ ((nei‘𝐽)‘{𝑝})))

Distinct variable groups:   𝐽,𝑝   𝑁,𝑝   𝑆,𝑝   𝑋,𝑝

Proof of Theorem neips

Dummy variables 𝑔 ℎ 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snssi 4746	. . . . . 6 ⊢ (𝑝 ∈ 𝑆 → {𝑝} ⊆ 𝑆)
2		neiss 23407	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ {𝑝} ⊆ 𝑆) → 𝑁 ∈ ((nei‘𝐽)‘{𝑝}))
3	1, 2	syl3an3 1183	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑝 ∈ 𝑆) → 𝑁 ∈ ((nei‘𝐽)‘{𝑝}))
4	3	3exp 1137	. . . 4 ⊢ (𝐽 ∈ Top → (𝑁 ∈ ((nei‘𝐽)‘𝑆) → (𝑝 ∈ 𝑆 → 𝑁 ∈ ((nei‘𝐽)‘{𝑝}))))
5	4	ralrimdv 3161	. . 3 ⊢ (𝐽 ∈ Top → (𝑁 ∈ ((nei‘𝐽)‘𝑆) → ∀𝑝 ∈ 𝑆 𝑁 ∈ ((nei‘𝐽)‘{𝑝})))
6	5	3ad2ant1 1151	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) → ∀𝑝 ∈ 𝑆 𝑁 ∈ ((nei‘𝐽)‘{𝑝})))
7		r19.28zv 4462	. . . . 5 ⊢ (𝑆 ≠ ∅ → (∀𝑝 ∈ 𝑆 (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)) ↔ (𝑁 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
8	7	3ad2ant3 1153	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (∀𝑝 ∈ 𝑆 (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)) ↔ (𝑁 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
9		ssrab2 4028	. . . . . . . . . 10 ⊢ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ⊆ 𝐽
10		uniopn 23195	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ⊆ 𝐽) → ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ∈ 𝐽)
11	9, 10	mpan2 704	. . . . . . . . 9 ⊢ (𝐽 ∈ Top → ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ∈ 𝐽)
12	11	ad2antrr 739	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ ∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)) → ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ∈ 𝐽)
13		sseq1 3956	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑔 → (𝑣 ⊆ 𝑁 ↔ 𝑔 ⊆ 𝑁))
14	13	elrab 3645	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ↔ (𝑔 ∈ 𝐽 ∧ 𝑔 ⊆ 𝑁))
15		elunii 4872	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∈ 𝑔 ∧ 𝑔 ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁}) → 𝑝 ∈ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁})
16	14, 15	sylan2br 607	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ 𝑔 ∧ (𝑔 ∈ 𝐽 ∧ 𝑔 ⊆ 𝑁)) → 𝑝 ∈ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁})
17	16	an12s 662	. . . . . . . . . . . . 13 ⊢ ((𝑔 ∈ 𝐽 ∧ (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)) → 𝑝 ∈ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁})
18	17	rexlimiva 3156	. . . . . . . . . . . 12 ⊢ (∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁) → 𝑝 ∈ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁})
19	18	ralimi 3100	. . . . . . . . . . 11 ⊢ (∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁) → ∀𝑝 ∈ 𝑆 𝑝 ∈ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁})
20		dfss3 3920	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ↔ ∀𝑝 ∈ 𝑆 𝑝 ∈ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁})
21	19, 20	sylibr 237	. . . . . . . . . 10 ⊢ (∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁) → 𝑆 ⊆ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁})
22	21	adantl 487	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ ∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)) → 𝑆 ⊆ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁})
23		unissb 4901	. . . . . . . . . 10 ⊢ (∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ⊆ 𝑁 ↔ ∀ℎ ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁}ℎ ⊆ 𝑁)
24		sseq1 3956	. . . . . . . . . . . 12 ⊢ (𝑣 = ℎ → (𝑣 ⊆ 𝑁 ↔ ℎ ⊆ 𝑁))
25	24	elrab 3645	. . . . . . . . . . 11 ⊢ (ℎ ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ↔ (ℎ ∈ 𝐽 ∧ ℎ ⊆ 𝑁))
26	25	simprbi 503	. . . . . . . . . 10 ⊢ (ℎ ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} → ℎ ⊆ 𝑁)
27	23, 26	mprgbir 3084	. . . . . . . . 9 ⊢ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ⊆ 𝑁
28	22, 27	jctir 530	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ ∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)) → (𝑆 ⊆ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ∧ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ⊆ 𝑁))
29		sseq2 3957	. . . . . . . . . 10 ⊢ (ℎ = ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} → (𝑆 ⊆ ℎ ↔ 𝑆 ⊆ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁}))
30		sseq1 3956	. . . . . . . . . 10 ⊢ (ℎ = ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} → (ℎ ⊆ 𝑁 ↔ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ⊆ 𝑁))
31	29, 30	anbi12d 644	. . . . . . . . 9 ⊢ (ℎ = ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} → ((𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁) ↔ (𝑆 ⊆ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ∧ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ⊆ 𝑁)))
32	31	rspcev 3577	. . . . . . . 8 ⊢ ((∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ∈ 𝐽 ∧ (𝑆 ⊆ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ∧ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ⊆ 𝑁)) → ∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁))
33	12, 28, 32	syl2anc 596	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ ∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)) → ∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁))
34	33	ex 418	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁) → ∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁)))
35	34	anim2d 624	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑁 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)) → (𝑁 ⊆ 𝑋 ∧ ∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁))))
36	35	3adant3 1150	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → ((𝑁 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)) → (𝑁 ⊆ 𝑋 ∧ ∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁))))
37	8, 36	sylbid 243	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (∀𝑝 ∈ 𝑆 (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)) → (𝑁 ⊆ 𝑋 ∧ ∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁))))
38		ssel2 3926	. . . . . . 7 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑝 ∈ 𝑆) → 𝑝 ∈ 𝑋)
39		neips.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
40	39	isneip 23403	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘{𝑝}) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
41	38, 40	sylan2 605	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ⊆ 𝑋 ∧ 𝑝 ∈ 𝑆)) → (𝑁 ∈ ((nei‘𝐽)‘{𝑝}) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
42	41	anassrs 473	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑝 ∈ 𝑆) → (𝑁 ∈ ((nei‘𝐽)‘{𝑝}) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
43	42	ralbidva 3184	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∀𝑝 ∈ 𝑆 𝑁 ∈ ((nei‘𝐽)‘{𝑝}) ↔ ∀𝑝 ∈ 𝑆 (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
44	43	3adant3 1150	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (∀𝑝 ∈ 𝑆 𝑁 ∈ ((nei‘𝐽)‘{𝑝}) ↔ ∀𝑝 ∈ 𝑆 (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
45	39	isnei 23401	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁))))
46	45	3adant3 1150	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁))))
47	37, 44, 46	3imtr4d 297	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (∀𝑝 ∈ 𝑆 𝑁 ∈ ((nei‘𝐽)‘{𝑝}) → 𝑁 ∈ ((nei‘𝐽)‘𝑆)))
48	6, 47	impbid 215	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ ∀𝑝 ∈ 𝑆 𝑁 ∈ ((nei‘𝐽)‘{𝑝})))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  {crab 3413   ⊆ wss 3899  ∅c0 4279  {csn 4584  ∪ cuni 4867  ‘cfv 6531  Topctop 23191  neicnei 23395

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-top 23192  df-nei 23396

This theorem is used by:  utop2nei  24549