Theorem neips 23049

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 4784	. . . . . 6 ⊢ (𝑝 ∈ 𝑆 → {𝑝} ⊆ 𝑆)
2		neiss 23045	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ {𝑝} ⊆ 𝑆) → 𝑁 ∈ ((nei‘𝐽)‘{𝑝}))
3	1, 2	syl3an3 1165	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑝 ∈ 𝑆) → 𝑁 ∈ ((nei‘𝐽)‘{𝑝}))
4	3	3exp 1119	. . . 4 ⊢ (𝐽 ∈ Top → (𝑁 ∈ ((nei‘𝐽)‘𝑆) → (𝑝 ∈ 𝑆 → 𝑁 ∈ ((nei‘𝐽)‘{𝑝}))))
5	4	ralrimdv 3138	. . 3 ⊢ (𝐽 ∈ Top → (𝑁 ∈ ((nei‘𝐽)‘𝑆) → ∀𝑝 ∈ 𝑆 𝑁 ∈ ((nei‘𝐽)‘{𝑝})))
6	5	3ad2ant1 1133	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) → ∀𝑝 ∈ 𝑆 𝑁 ∈ ((nei‘𝐽)‘{𝑝})))
7		r19.28zv 4476	. . . . 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 1135	. . . 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 4055	. . . . . . . . . 10 ⊢ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ⊆ 𝐽
10		uniopn 22833	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ⊆ 𝐽) → ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ∈ 𝐽)
11	9, 10	mpan2 691	. . . . . . . . 9 ⊢ (𝐽 ∈ Top → ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ∈ 𝐽)
12	11	ad2antrr 726	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ ∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)) → ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ∈ 𝐽)
13		sseq1 3984	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = 𝑔 → (𝑣 ⊆ 𝑁 ↔ 𝑔 ⊆ 𝑁))
14	13	elrab 3671	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ↔ (𝑔 ∈ 𝐽 ∧ 𝑔 ⊆ 𝑁))
15		elunii 4888	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∈ 𝑔 ∧ 𝑔 ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁}) → 𝑝 ∈ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁})
16	14, 15	sylan2br 595	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ∈ 𝑔 ∧ (𝑔 ∈ 𝐽 ∧ 𝑔 ⊆ 𝑁)) → 𝑝 ∈ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁})
17	16	an12s 649	. . . . . . . . . . . . 13 ⊢ ((𝑔 ∈ 𝐽 ∧ (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)) → 𝑝 ∈ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁})
18	17	rexlimiva 3133	. . . . . . . . . . . 12 ⊢ (∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁) → 𝑝 ∈ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁})
19	18	ralimi 3073	. . . . . . . . . . 11 ⊢ (∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁) → ∀𝑝 ∈ 𝑆 𝑝 ∈ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁})
20		dfss3 3947	. . . . . . . . . . 11 ⊢ (𝑆 ⊆ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ↔ ∀𝑝 ∈ 𝑆 𝑝 ∈ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁})
21	19, 20	sylibr 234	. . . . . . . . . 10 ⊢ (∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁) → 𝑆 ⊆ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁})
22	21	adantl 481	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ ∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)) → 𝑆 ⊆ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁})
23		unissb 4915	. . . . . . . . . 10 ⊢ (∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ⊆ 𝑁 ↔ ∀ℎ ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁}ℎ ⊆ 𝑁)
24		sseq1 3984	. . . . . . . . . . . 12 ⊢ (𝑣 = ℎ → (𝑣 ⊆ 𝑁 ↔ ℎ ⊆ 𝑁))
25	24	elrab 3671	. . . . . . . . . . 11 ⊢ (ℎ ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ↔ (ℎ ∈ 𝐽 ∧ ℎ ⊆ 𝑁))
26	25	simprbi 496	. . . . . . . . . 10 ⊢ (ℎ ∈ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} → ℎ ⊆ 𝑁)
27	23, 26	mprgbir 3058	. . . . . . . . 9 ⊢ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ⊆ 𝑁
28	22, 27	jctir 520	. . . . . . . 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 3985	. . . . . . . . . 10 ⊢ (ℎ = ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} → (𝑆 ⊆ ℎ ↔ 𝑆 ⊆ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁}))
30		sseq1 3984	. . . . . . . . . 10 ⊢ (ℎ = ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} → (ℎ ⊆ 𝑁 ↔ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ⊆ 𝑁))
31	29, 30	anbi12d 632	. . . . . . . . 9 ⊢ (ℎ = ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} → ((𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁) ↔ (𝑆 ⊆ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ∧ ∪ {𝑣 ∈ 𝐽 ∣ 𝑣 ⊆ 𝑁} ⊆ 𝑁)))
32	31	rspcev 3601	. . . . . . . 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 584	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ ∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁)) → ∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁))
34	33	ex 412	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∀𝑝 ∈ 𝑆 ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁) → ∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁)))
35	34	anim2d 612	. . . . 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 1132	. . . 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 240	. . 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 3953	. . . . . . 7 ⊢ ((𝑆 ⊆ 𝑋 ∧ 𝑝 ∈ 𝑆) → 𝑝 ∈ 𝑋)
39		neips.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
40	39	isneip 23041	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘{𝑝}) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
41	38, 40	sylan2 593	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ⊆ 𝑋 ∧ 𝑝 ∈ 𝑆)) → (𝑁 ∈ ((nei‘𝐽)‘{𝑝}) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
42	41	anassrs 467	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑝 ∈ 𝑆) → (𝑁 ∈ ((nei‘𝐽)‘{𝑝}) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
43	42	ralbidva 3161	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∀𝑝 ∈ 𝑆 𝑁 ∈ ((nei‘𝐽)‘{𝑝}) ↔ ∀𝑝 ∈ 𝑆 (𝑁 ⊆ 𝑋 ∧ ∃𝑔 ∈ 𝐽 (𝑝 ∈ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
44	43	3adant3 1132	. . 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 23039	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁))))
46	45	3adant3 1132	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁 ⊆ 𝑋 ∧ ∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁))))
47	37, 44, 46	3imtr4d 294	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (∀𝑝 ∈ 𝑆 𝑁 ∈ ((nei‘𝐽)‘{𝑝}) → 𝑁 ∈ ((nei‘𝐽)‘𝑆)))
48	6, 47	impbid 212	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑆 ≠ ∅) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ ∀𝑝 ∈ 𝑆 𝑁 ∈ ((nei‘𝐽)‘{𝑝})))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  {crab 3415   ⊆ wss 3926  ∅c0 4308  {csn 4601  ∪ cuni 4883  ‘cfv 6530  Topctop 22829  neicnei 23033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-top 22830  df-nei 23034

This theorem is referenced by:  utop2nei  24187