Theorem innei 23133

Description: The intersection of two neighborhoods of a set is also a neighborhood of the set. Generalization to subsets of Property V_ii of [BourbakiTop1] p. I.3 for binary intersections. (Contributed by FL, 28-Sep-2006.)

Assertion

Ref	Expression
innei	⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁 ∩ 𝑀) ∈ ((nei‘𝐽)‘𝑆))

Proof of Theorem innei

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

Step	Hyp	Ref	Expression
1		eqid 2737	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	neii1 23114	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁 ⊆ ∪ 𝐽)
3		ssinss1 4246	. . . 4 ⊢ (𝑁 ⊆ ∪ 𝐽 → (𝑁 ∩ 𝑀) ⊆ ∪ 𝐽)
4	2, 3	syl 17	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁 ∩ 𝑀) ⊆ ∪ 𝐽)
5	4	3adant3 1133	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁 ∩ 𝑀) ⊆ ∪ 𝐽)
6		neii2 23116	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁))
7		neii2 23116	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑀))
8	6, 7	anim12dan 619	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆))) → (∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁) ∧ ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑀)))
9		inopn 22905	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ ℎ ∈ 𝐽 ∧ 𝑣 ∈ 𝐽) → (ℎ ∩ 𝑣) ∈ 𝐽)
10	9	3expa 1119	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ ℎ ∈ 𝐽) ∧ 𝑣 ∈ 𝐽) → (ℎ ∩ 𝑣) ∈ 𝐽)
11		ssin 4239	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ ℎ ∧ 𝑆 ⊆ 𝑣) ↔ 𝑆 ⊆ (ℎ ∩ 𝑣))
12	11	biimpi 216	. . . . . . . . . . . 12 ⊢ ((𝑆 ⊆ ℎ ∧ 𝑆 ⊆ 𝑣) → 𝑆 ⊆ (ℎ ∩ 𝑣))
13		ss2in 4245	. . . . . . . . . . . 12 ⊢ ((ℎ ⊆ 𝑁 ∧ 𝑣 ⊆ 𝑀) → (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀))
14	12, 13	anim12i 613	. . . . . . . . . . 11 ⊢ (((𝑆 ⊆ ℎ ∧ 𝑆 ⊆ 𝑣) ∧ (ℎ ⊆ 𝑁 ∧ 𝑣 ⊆ 𝑀)) → (𝑆 ⊆ (ℎ ∩ 𝑣) ∧ (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀)))
15	14	an4s 660	. . . . . . . . . 10 ⊢ (((𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁) ∧ (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑀)) → (𝑆 ⊆ (ℎ ∩ 𝑣) ∧ (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀)))
16		sseq2 4010	. . . . . . . . . . . 12 ⊢ (𝑔 = (ℎ ∩ 𝑣) → (𝑆 ⊆ 𝑔 ↔ 𝑆 ⊆ (ℎ ∩ 𝑣)))
17		sseq1 4009	. . . . . . . . . . . 12 ⊢ (𝑔 = (ℎ ∩ 𝑣) → (𝑔 ⊆ (𝑁 ∩ 𝑀) ↔ (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀)))
18	16, 17	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑔 = (ℎ ∩ 𝑣) → ((𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)) ↔ (𝑆 ⊆ (ℎ ∩ 𝑣) ∧ (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀))))
19	18	rspcev 3622	. . . . . . . . . 10 ⊢ (((ℎ ∩ 𝑣) ∈ 𝐽 ∧ (𝑆 ⊆ (ℎ ∩ 𝑣) ∧ (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀))) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))
20	10, 15, 19	syl2an 596	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Top ∧ ℎ ∈ 𝐽) ∧ 𝑣 ∈ 𝐽) ∧ ((𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁) ∧ (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑀))) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))
21	20	expr 456	. . . . . . . 8 ⊢ ((((𝐽 ∈ Top ∧ ℎ ∈ 𝐽) ∧ 𝑣 ∈ 𝐽) ∧ (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁)) → ((𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑀) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀))))
22	21	an32s 652	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ ℎ ∈ 𝐽) ∧ (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁)) ∧ 𝑣 ∈ 𝐽) → ((𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑀) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀))))
23	22	rexlimdva 3155	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ ℎ ∈ 𝐽) ∧ (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁)) → (∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑀) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀))))
24	23	rexlimdva2 3157	. . . . 5 ⊢ (𝐽 ∈ Top → (∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁) → (∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑀) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))))
25	24	imp32 418	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁) ∧ ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑀))) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))
26	8, 25	syldan 591	. . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆))) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))
27	26	3impb 1115	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))
28	1	neiss2 23109	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ ∪ 𝐽)
29	1	isnei 23111	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((𝑁 ∩ 𝑀) ∈ ((nei‘𝐽)‘𝑆) ↔ ((𝑁 ∩ 𝑀) ⊆ ∪ 𝐽 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))))
30	28, 29	syldan 591	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ((𝑁 ∩ 𝑀) ∈ ((nei‘𝐽)‘𝑆) ↔ ((𝑁 ∩ 𝑀) ⊆ ∪ 𝐽 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))))
31	30	3adant3 1133	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → ((𝑁 ∩ 𝑀) ∈ ((nei‘𝐽)‘𝑆) ↔ ((𝑁 ∩ 𝑀) ⊆ ∪ 𝐽 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))))
32	5, 27, 31	mpbir2and 713	1 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁 ∩ 𝑀) ∈ ((nei‘𝐽)‘𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∃wrex 3070   ∩ cin 3950   ⊆ wss 3951  ∪ cuni 4907  ‘cfv 6561  Topctop 22899  neicnei 23105

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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-top 22900  df-nei 23106

This theorem is referenced by:  neifil  23888  neificl  37760