Theorem neiin 34258

Description: Two neighborhoods intersect to form a neighborhood of the intersection. (Contributed by Jeff Hankins, 31-Aug-2009.)

Assertion

Ref	Expression
neiin	⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝑀 ∩ 𝑁) ∈ ((nei‘𝐽)‘(𝐴 ∩ 𝐵)))

Proof of Theorem neiin

Step	Hyp	Ref	Expression
1		simpr 488	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝑀 ∈ ((nei‘𝐽)‘𝐴))
2		simpl 486	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝐽 ∈ Top)
3		eqid 2737	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	neiss2 21998	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝐴 ⊆ ∪ 𝐽)
5	3	neii1 22003	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝑀 ⊆ ∪ 𝐽)
6	3	neiint 22001	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽 ∧ 𝑀 ⊆ ∪ 𝐽) → (𝑀 ∈ ((nei‘𝐽)‘𝐴) ↔ 𝐴 ⊆ ((int‘𝐽)‘𝑀)))
7	2, 4, 5, 6	syl3anc 1373	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝑀 ∈ ((nei‘𝐽)‘𝐴) ↔ 𝐴 ⊆ ((int‘𝐽)‘𝑀)))
8	1, 7	mpbid 235	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝐴 ⊆ ((int‘𝐽)‘𝑀))
9		ssinss1 4152	. . . . . 6 ⊢ (𝐴 ⊆ ((int‘𝐽)‘𝑀) → (𝐴 ∩ 𝐵) ⊆ ((int‘𝐽)‘𝑀))
10	8, 9	syl 17	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝐴 ∩ 𝐵) ⊆ ((int‘𝐽)‘𝑀))
11	10	3adant3 1134	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴 ∩ 𝐵) ⊆ ((int‘𝐽)‘𝑀))
12		inss2 4144	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
13		simpr 488	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑁 ∈ ((nei‘𝐽)‘𝐵))
14		simpl 486	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐽 ∈ Top)
15	3	neiss2 21998	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐵 ⊆ ∪ 𝐽)
16	3	neii1 22003	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑁 ⊆ ∪ 𝐽)
17	3	neiint 22001	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ ∪ 𝐽 ∧ 𝑁 ⊆ ∪ 𝐽) → (𝑁 ∈ ((nei‘𝐽)‘𝐵) ↔ 𝐵 ⊆ ((int‘𝐽)‘𝑁)))
18	14, 15, 16, 17	syl3anc 1373	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝑁 ∈ ((nei‘𝐽)‘𝐵) ↔ 𝐵 ⊆ ((int‘𝐽)‘𝑁)))
19	13, 18	mpbid 235	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐵 ⊆ ((int‘𝐽)‘𝑁))
20	19	3adant2 1133	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐵 ⊆ ((int‘𝐽)‘𝑁))
21	12, 20	sstrid 3912	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴 ∩ 𝐵) ⊆ ((int‘𝐽)‘𝑁))
22	11, 21	ssind 4147	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴 ∩ 𝐵) ⊆ (((int‘𝐽)‘𝑀) ∩ ((int‘𝐽)‘𝑁)))
23		simp1 1138	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐽 ∈ Top)
24	5	3adant3 1134	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑀 ⊆ ∪ 𝐽)
25	16	3adant2 1133	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑁 ⊆ ∪ 𝐽)
26	3	ntrin 21958	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ⊆ ∪ 𝐽 ∧ 𝑁 ⊆ ∪ 𝐽) → ((int‘𝐽)‘(𝑀 ∩ 𝑁)) = (((int‘𝐽)‘𝑀) ∩ ((int‘𝐽)‘𝑁)))
27	23, 24, 25, 26	syl3anc 1373	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → ((int‘𝐽)‘(𝑀 ∩ 𝑁)) = (((int‘𝐽)‘𝑀) ∩ ((int‘𝐽)‘𝑁)))
28	22, 27	sseqtrrd 3942	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴 ∩ 𝐵) ⊆ ((int‘𝐽)‘(𝑀 ∩ 𝑁)))
29		ssinss1 4152	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝐽 → (𝐴 ∩ 𝐵) ⊆ ∪ 𝐽)
30	4, 29	syl 17	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝐴 ∩ 𝐵) ⊆ ∪ 𝐽)
31		ssinss1 4152	. . . . 5 ⊢ (𝑀 ⊆ ∪ 𝐽 → (𝑀 ∩ 𝑁) ⊆ ∪ 𝐽)
32	5, 31	syl 17	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝑀 ∩ 𝑁) ⊆ ∪ 𝐽)
33	3	neiint 22001	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝐴 ∩ 𝐵) ⊆ ∪ 𝐽 ∧ (𝑀 ∩ 𝑁) ⊆ ∪ 𝐽) → ((𝑀 ∩ 𝑁) ∈ ((nei‘𝐽)‘(𝐴 ∩ 𝐵)) ↔ (𝐴 ∩ 𝐵) ⊆ ((int‘𝐽)‘(𝑀 ∩ 𝑁))))
34	2, 30, 32, 33	syl3anc 1373	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → ((𝑀 ∩ 𝑁) ∈ ((nei‘𝐽)‘(𝐴 ∩ 𝐵)) ↔ (𝐴 ∩ 𝐵) ⊆ ((int‘𝐽)‘(𝑀 ∩ 𝑁))))
35	34	3adant3 1134	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → ((𝑀 ∩ 𝑁) ∈ ((nei‘𝐽)‘(𝐴 ∩ 𝐵)) ↔ (𝐴 ∩ 𝐵) ⊆ ((int‘𝐽)‘(𝑀 ∩ 𝑁))))
36	28, 35	mpbird 260	1 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝑀 ∩ 𝑁) ∈ ((nei‘𝐽)‘(𝐴 ∩ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110   ∩ cin 3865   ⊆ wss 3866  ∪ cuni 4819  ‘cfv 6380  Topctop 21790  intcnt 21914  neicnei 21994

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-top 21791  df-cld 21916  df-ntr 21917  df-cls 21918  df-nei 21995

This theorem is referenced by: (None)