Theorem innei 21735

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 2823	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	neii1 21716	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁 ⊆ ∪ 𝐽)
3		ssinss1 4216	. . . 4 ⊢ (𝑁 ⊆ ∪ 𝐽 → (𝑁 ∩ 𝑀) ⊆ ∪ 𝐽)
4	2, 3	syl 17	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁 ∩ 𝑀) ⊆ ∪ 𝐽)
5	4	3adant3 1128	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁 ∩ 𝑀) ⊆ ∪ 𝐽)
6		neii2 21718	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁))
7		neii2 21718	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑀))
8	6, 7	anim12dan 620	. . . 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 21509	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ ℎ ∈ 𝐽 ∧ 𝑣 ∈ 𝐽) → (ℎ ∩ 𝑣) ∈ 𝐽)
10	9	3expa 1114	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ ℎ ∈ 𝐽) ∧ 𝑣 ∈ 𝐽) → (ℎ ∩ 𝑣) ∈ 𝐽)
11		ssin 4209	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ ℎ ∧ 𝑆 ⊆ 𝑣) ↔ 𝑆 ⊆ (ℎ ∩ 𝑣))
12	11	biimpi 218	. . . . . . . . . . . 12 ⊢ ((𝑆 ⊆ ℎ ∧ 𝑆 ⊆ 𝑣) → 𝑆 ⊆ (ℎ ∩ 𝑣))
13		ss2in 4215	. . . . . . . . . . . 12 ⊢ ((ℎ ⊆ 𝑁 ∧ 𝑣 ⊆ 𝑀) → (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀))
14	12, 13	anim12i 614	. . . . . . . . . . 11 ⊢ (((𝑆 ⊆ ℎ ∧ 𝑆 ⊆ 𝑣) ∧ (ℎ ⊆ 𝑁 ∧ 𝑣 ⊆ 𝑀)) → (𝑆 ⊆ (ℎ ∩ 𝑣) ∧ (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀)))
15	14	an4s 658	. . . . . . . . . 10 ⊢ (((𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁) ∧ (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑀)) → (𝑆 ⊆ (ℎ ∩ 𝑣) ∧ (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀)))
16		sseq2 3995	. . . . . . . . . . . 12 ⊢ (𝑔 = (ℎ ∩ 𝑣) → (𝑆 ⊆ 𝑔 ↔ 𝑆 ⊆ (ℎ ∩ 𝑣)))
17		sseq1 3994	. . . . . . . . . . . 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 3625	. . . . . . . . . 10 ⊢ (((ℎ ∩ 𝑣) ∈ 𝐽 ∧ (𝑆 ⊆ (ℎ ∩ 𝑣) ∧ (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀))) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))
20	10, 15, 19	syl2an 597	. . . . . . . . 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 459	. . . . . . . 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 650	. . . . . . 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 3286	. . . . . 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 3289	. . . . 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 421	. . . 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 593	. . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆))) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))
27	26	3impb 1111	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))
28	1	neiss2 21711	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ ∪ 𝐽)
29	1	isnei 21713	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((𝑁 ∩ 𝑀) ∈ ((nei‘𝐽)‘𝑆) ↔ ((𝑁 ∩ 𝑀) ⊆ ∪ 𝐽 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))))
30	28, 29	syldan 593	. . 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 1128	. 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 711	1 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁 ∩ 𝑀) ∈ ((nei‘𝐽)‘𝑆))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∃wrex 3141   ∩ cin 3937   ⊆ wss 3938  ∪ cuni 4840  ‘cfv 6357  Topctop 21503  neicnei 21707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-top 21504  df-nei 21708

This theorem is referenced by:  neifil  22490  neificl  35030