Theorem neiin 36397

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 484	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝑀 ∈ ((nei‘𝐽)‘𝐴))
2		simpl 482	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝐽 ∈ Top)
3		eqid 2733	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	neiss2 23017	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝐴 ⊆ ∪ 𝐽)
5	3	neii1 23022	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝑀 ⊆ ∪ 𝐽)
6	3	neiint 23020	. . . . . . . 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 232	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝐴 ⊆ ((int‘𝐽)‘𝑀))
9		ssinss1 4195	. . . . . 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 1132	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴 ∩ 𝐵) ⊆ ((int‘𝐽)‘𝑀))
12		inss2 4187	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
13		simpr 484	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑁 ∈ ((nei‘𝐽)‘𝐵))
14		simpl 482	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐽 ∈ Top)
15	3	neiss2 23017	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐵 ⊆ ∪ 𝐽)
16	3	neii1 23022	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑁 ⊆ ∪ 𝐽)
17	3	neiint 23020	. . . . . . . 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 232	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐵 ⊆ ((int‘𝐽)‘𝑁))
20	19	3adant2 1131	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐵 ⊆ ((int‘𝐽)‘𝑁))
21	12, 20	sstrid 3942	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴 ∩ 𝐵) ⊆ ((int‘𝐽)‘𝑁))
22	11, 21	ssind 4190	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴 ∩ 𝐵) ⊆ (((int‘𝐽)‘𝑀) ∩ ((int‘𝐽)‘𝑁)))
23		simp1 1136	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐽 ∈ Top)
24	5	3adant3 1132	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑀 ⊆ ∪ 𝐽)
25	16	3adant2 1131	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑁 ⊆ ∪ 𝐽)
26	3	ntrin 22977	. . . 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 3968	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝐴 ∩ 𝐵) ⊆ ((int‘𝐽)‘(𝑀 ∩ 𝑁)))
29		ssinss1 4195	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝐽 → (𝐴 ∩ 𝐵) ⊆ ∪ 𝐽)
30	4, 29	syl 17	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝐴 ∩ 𝐵) ⊆ ∪ 𝐽)
31		ssinss1 4195	. . . . 5 ⊢ (𝑀 ⊆ ∪ 𝐽 → (𝑀 ∩ 𝑁) ⊆ ∪ 𝐽)
32	5, 31	syl 17	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝑀 ∩ 𝑁) ⊆ ∪ 𝐽)
33	3	neiint 23020	. . . 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 1132	. 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 257	1 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → (𝑀 ∩ 𝑁) ∈ ((nei‘𝐽)‘(𝐴 ∩ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ∩ cin 3897   ⊆ wss 3898  ∪ cuni 4858  ‘cfv 6486  Topctop 22809  intcnt 22933  neicnei 23013

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-iin 4944  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-top 22810  df-cld 22935  df-ntr 22936  df-cls 22937  df-nei 23014

This theorem is referenced by: (None)