Theorem innei 23154

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 2740	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	neii1 23135	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁 ⊆ ∪ 𝐽)
3		ssinss1 4267	. . . 4 ⊢ (𝑁 ⊆ ∪ 𝐽 → (𝑁 ∩ 𝑀) ⊆ ∪ 𝐽)
4	2, 3	syl 17	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁 ∩ 𝑀) ⊆ ∪ 𝐽)
5	4	3adant3 1132	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁 ∩ 𝑀) ⊆ ∪ 𝐽)
6		neii2 23137	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁))
7		neii2 23137	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑀))
8	6, 7	anim12dan 618	. . . 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 22926	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ ℎ ∈ 𝐽 ∧ 𝑣 ∈ 𝐽) → (ℎ ∩ 𝑣) ∈ 𝐽)
10	9	3expa 1118	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ ℎ ∈ 𝐽) ∧ 𝑣 ∈ 𝐽) → (ℎ ∩ 𝑣) ∈ 𝐽)
11		ssin 4260	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ ℎ ∧ 𝑆 ⊆ 𝑣) ↔ 𝑆 ⊆ (ℎ ∩ 𝑣))
12	11	biimpi 216	. . . . . . . . . . . 12 ⊢ ((𝑆 ⊆ ℎ ∧ 𝑆 ⊆ 𝑣) → 𝑆 ⊆ (ℎ ∩ 𝑣))
13		ss2in 4266	. . . . . . . . . . . 12 ⊢ ((ℎ ⊆ 𝑁 ∧ 𝑣 ⊆ 𝑀) → (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀))
14	12, 13	anim12i 612	. . . . . . . . . . 11 ⊢ (((𝑆 ⊆ ℎ ∧ 𝑆 ⊆ 𝑣) ∧ (ℎ ⊆ 𝑁 ∧ 𝑣 ⊆ 𝑀)) → (𝑆 ⊆ (ℎ ∩ 𝑣) ∧ (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀)))
15	14	an4s 659	. . . . . . . . . 10 ⊢ (((𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁) ∧ (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑀)) → (𝑆 ⊆ (ℎ ∩ 𝑣) ∧ (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀)))
16		sseq2 4035	. . . . . . . . . . . 12 ⊢ (𝑔 = (ℎ ∩ 𝑣) → (𝑆 ⊆ 𝑔 ↔ 𝑆 ⊆ (ℎ ∩ 𝑣)))
17		sseq1 4034	. . . . . . . . . . . 12 ⊢ (𝑔 = (ℎ ∩ 𝑣) → (𝑔 ⊆ (𝑁 ∩ 𝑀) ↔ (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀)))
18	16, 17	anbi12d 631	. . . . . . . . . . 11 ⊢ (𝑔 = (ℎ ∩ 𝑣) → ((𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)) ↔ (𝑆 ⊆ (ℎ ∩ 𝑣) ∧ (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀))))
19	18	rspcev 3635	. . . . . . . . . 10 ⊢ (((ℎ ∩ 𝑣) ∈ 𝐽 ∧ (𝑆 ⊆ (ℎ ∩ 𝑣) ∧ (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀))) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))
20	10, 15, 19	syl2an 595	. . . . . . . . 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 651	. . . . . . 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 3161	. . . . . 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 3163	. . . . 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 590	. . 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 23130	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ ∪ 𝐽)
29	1	isnei 23132	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((𝑁 ∩ 𝑀) ∈ ((nei‘𝐽)‘𝑆) ↔ ((𝑁 ∩ 𝑀) ⊆ ∪ 𝐽 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))))
30	28, 29	syldan 590	. . 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 1132	. 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 712	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 1537   ∈ wcel 2108  ∃wrex 3076   ∩ cin 3975   ⊆ wss 3976  ∪ cuni 4931  ‘cfv 6573  Topctop 22920  neicnei 23126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-top 22921  df-nei 23127

This theorem is referenced by:  neifil  23909  neificl  37713