Theorem innei 20929

Description: The intersection of two neighborhoods of a set is also a neighborhood of the set. Proposition Vii of [BourbakiTop1] p. I.3 . (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 2622	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	neii1 20910	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁 ⊆ ∪ 𝐽)
3		ssinss1 3841	. . . 4 ⊢ (𝑁 ⊆ ∪ 𝐽 → (𝑁 ∩ 𝑀) ⊆ ∪ 𝐽)
4	2, 3	syl 17	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁 ∩ 𝑀) ⊆ ∪ 𝐽)
5	4	3adant3 1081	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁 ∩ 𝑀) ⊆ ∪ 𝐽)
6		neii2 20912	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁))
7		neii2 20912	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑀))
8	6, 7	anim12dan 882	. . . 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 20704	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ ℎ ∈ 𝐽 ∧ 𝑣 ∈ 𝐽) → (ℎ ∩ 𝑣) ∈ 𝐽)
10	9	3expa 1265	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ ℎ ∈ 𝐽) ∧ 𝑣 ∈ 𝐽) → (ℎ ∩ 𝑣) ∈ 𝐽)
11		ssin 3835	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ⊆ ℎ ∧ 𝑆 ⊆ 𝑣) ↔ 𝑆 ⊆ (ℎ ∩ 𝑣))
12	11	biimpi 206	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ ℎ ∧ 𝑆 ⊆ 𝑣) → 𝑆 ⊆ (ℎ ∩ 𝑣))
13		ss2in 3840	. . . . . . . . . . . . 13 ⊢ ((ℎ ⊆ 𝑁 ∧ 𝑣 ⊆ 𝑀) → (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀))
14	12, 13	anim12i 590	. . . . . . . . . . . 12 ⊢ (((𝑆 ⊆ ℎ ∧ 𝑆 ⊆ 𝑣) ∧ (ℎ ⊆ 𝑁 ∧ 𝑣 ⊆ 𝑀)) → (𝑆 ⊆ (ℎ ∩ 𝑣) ∧ (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀)))
15	14	an4s 869	. . . . . . . . . . 11 ⊢ (((𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁) ∧ (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑀)) → (𝑆 ⊆ (ℎ ∩ 𝑣) ∧ (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀)))
16		sseq2 3627	. . . . . . . . . . . . 13 ⊢ (𝑔 = (ℎ ∩ 𝑣) → (𝑆 ⊆ 𝑔 ↔ 𝑆 ⊆ (ℎ ∩ 𝑣)))
17		sseq1 3626	. . . . . . . . . . . . 13 ⊢ (𝑔 = (ℎ ∩ 𝑣) → (𝑔 ⊆ (𝑁 ∩ 𝑀) ↔ (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀)))
18	16, 17	anbi12d 747	. . . . . . . . . . . 12 ⊢ (𝑔 = (ℎ ∩ 𝑣) → ((𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)) ↔ (𝑆 ⊆ (ℎ ∩ 𝑣) ∧ (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀))))
19	18	rspcev 3309	. . . . . . . . . . 11 ⊢ (((ℎ ∩ 𝑣) ∈ 𝐽 ∧ (𝑆 ⊆ (ℎ ∩ 𝑣) ∧ (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀))) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))
20	10, 15, 19	syl2an 494	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ Top ∧ ℎ ∈ 𝐽) ∧ 𝑣 ∈ 𝐽) ∧ ((𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁) ∧ (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑀))) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))
21	20	expr 643	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Top ∧ ℎ ∈ 𝐽) ∧ 𝑣 ∈ 𝐽) ∧ (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁)) → ((𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑀) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀))))
22	21	an32s 846	. . . . . . . 8 ⊢ ((((𝐽 ∈ Top ∧ ℎ ∈ 𝐽) ∧ (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁)) ∧ 𝑣 ∈ 𝐽) → ((𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑀) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀))))
23	22	rexlimdva 3031	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ ℎ ∈ 𝐽) ∧ (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁)) → (∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑀) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀))))
24	23	ex 450	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ ℎ ∈ 𝐽) → ((𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁) → (∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑀) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))))
25	24	rexlimdva 3031	. . . . 5 ⊢ (𝐽 ∈ Top → (∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁) → (∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑀) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))))
26	25	imp32 449	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁) ∧ ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑀))) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))
27	8, 26	syldan 487	. . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆))) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))
28	27	3impb 1260	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))
29	1	neiss2 20905	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ ∪ 𝐽)
30	1	isnei 20907	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((𝑁 ∩ 𝑀) ∈ ((nei‘𝐽)‘𝑆) ↔ ((𝑁 ∩ 𝑀) ⊆ ∪ 𝐽 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))))
31	29, 30	syldan 487	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ((𝑁 ∩ 𝑀) ∈ ((nei‘𝐽)‘𝑆) ↔ ((𝑁 ∩ 𝑀) ⊆ ∪ 𝐽 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))))
32	31	3adant3 1081	. 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)))))
33	5, 28, 32	mpbir2and 957	1 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁 ∩ 𝑀) ∈ ((nei‘𝐽)‘𝑆))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∃wrex 2913   ∩ cin 3573   ⊆ wss 3574  ∪ cuni 4436  ‘cfv 5888  Topctop 20698  neicnei 20901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-top 20699  df-nei 20902

This theorem is referenced by:  neifil  21684  neificl  33549