Theorem neiin 36705

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 489	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝑀 ∈ ((nei‘𝐽)‘𝐴))
2		simpl 487	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝐽 ∈ Top)
3		eqid 2765	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	neiss2 23219	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝐴 ⊆ ∪ 𝐽)
5	3	neii1 23224	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝑀 ⊆ ∪ 𝐽)
6	3	neiint 23222	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽 ∧ 𝑀 ⊆ ∪ 𝐽) → (𝑀 ∈ ((nei‘𝐽)‘𝐴) ↔ 𝐴 ⊆ ((int‘𝐽)‘𝑀)))
7	2, 4, 5, 6	syl3anc 1394	. . . . . . 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 4200	. . . . . 6 ⊢ (𝐴 ⊆ ((int‘𝐽)‘𝑀) → (𝐴 ∩ 𝐵) ⊆ ((int‘𝐽)‘𝑀))
10	8, 9	syl 18	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝐴 ∩ 𝐵) ⊆ ((int‘𝐽)‘𝑀))
11	10	3adant3 1148	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴 ∩ 𝐵) ⊆ ((int‘𝐽)‘𝑀))
12		inss2 4192	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
13		simpr 489	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑁 ∈ ((nei‘𝐽)‘𝐵))
14		simpl 487	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐽 ∈ Top)
15	3	neiss2 23219	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐵 ⊆ ∪ 𝐽)
16	3	neii1 23224	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑁 ⊆ ∪ 𝐽)
17	3	neiint 23222	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ ∪ 𝐽 ∧ 𝑁 ⊆ ∪ 𝐽) → (𝑁 ∈ ((nei‘𝐽)‘𝐵) ↔ 𝐵 ⊆ ((int‘𝐽)‘𝑁)))
18	14, 15, 16, 17	syl3anc 1394	. . . . . . 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 1147	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐵 ⊆ ((int‘𝐽)‘𝑁))
21	12, 20	sstrid 3950	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴 ∩ 𝐵) ⊆ ((int‘𝐽)‘𝑁))
22	11, 21	ssind 4195	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴 ∩ 𝐵) ⊆ (((int‘𝐽)‘𝑀) ∩ ((int‘𝐽)‘𝑁)))
23		simp1 1152	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐽 ∈ Top)
24	5	3adant3 1148	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑀 ⊆ ∪ 𝐽)
25	16	3adant2 1147	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑁 ⊆ ∪ 𝐽)
26	3	ntrin 23179	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ⊆ ∪ 𝐽 ∧ 𝑁 ⊆ ∪ 𝐽) → ((int‘𝐽)‘(𝑀 ∩ 𝑁)) = (((int‘𝐽)‘𝑀) ∩ ((int‘𝐽)‘𝑁)))
27	23, 24, 25, 26	syl3anc 1394	. . 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 3976	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴 ∩ 𝐵) ⊆ ((int‘𝐽)‘(𝑀 ∩ 𝑁)))
29		ssinss1 4200	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝐽 → (𝐴 ∩ 𝐵) ⊆ ∪ 𝐽)
30	4, 29	syl 18	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝐴 ∩ 𝐵) ⊆ ∪ 𝐽)
31		ssinss1 4200	. . . . 5 ⊢ (𝑀 ⊆ ∪ 𝐽 → (𝑀 ∩ 𝑁) ⊆ ∪ 𝐽)
32	5, 31	syl 18	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝑀 ∩ 𝑁) ⊆ ∪ 𝐽)
33	3	neiint 23222	. . . 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 1394	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → ((𝑀 ∩ 𝑁) ∈ ((nei‘𝐽)‘(𝐴 ∩ 𝐵)) ↔ (𝐴 ∩ 𝐵) ⊆ ((int‘𝐽)‘(𝑀 ∩ 𝑁))))
35	34	3adant3 1148	. 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 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ∩ cin 3906   ⊆ wss 3907  ∪ cuni 4868  ‘cfv 6525  Topctop 23011  intcnt 23135  neicnei 23215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-iin 4955  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-top 23012  df-cld 23137  df-ntr 23138  df-cls 23139  df-nei 23216

This theorem is referenced by: (None)