Theorem neiin 36545

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 2737	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
4	3	neiss2 23057	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝐴 ⊆ ∪ 𝐽)
5	3	neii1 23062	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → 𝑀 ⊆ ∪ 𝐽)
6	3	neiint 23060	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽 ∧ 𝑀 ⊆ ∪ 𝐽) → (𝑀 ∈ ((nei‘𝐽)‘𝐴) ↔ 𝐴 ⊆ ((int‘𝐽)‘𝑀)))
7	2, 4, 5, 6	syl3anc 1374	. . . . . . 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 4200	. . . . . 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 1133	. . . 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 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 23057	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐵 ⊆ ∪ 𝐽)
16	3	neii1 23062	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑁 ⊆ ∪ 𝐽)
17	3	neiint 23060	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ⊆ ∪ 𝐽 ∧ 𝑁 ⊆ ∪ 𝐽) → (𝑁 ∈ ((nei‘𝐽)‘𝐵) ↔ 𝐵 ⊆ ((int‘𝐽)‘𝑁)))
18	14, 15, 16, 17	syl3anc 1374	. . . . . . 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 1132	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐵 ⊆ ((int‘𝐽)‘𝑁))
21	12, 20	sstrid 3947	. . . 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 1137	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝐽 ∈ Top)
24	5	3adant3 1133	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑀 ⊆ ∪ 𝐽)
25	16	3adant2 1132	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝐵)) → 𝑁 ⊆ ∪ 𝐽)
26	3	ntrin 23017	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ⊆ ∪ 𝐽 ∧ 𝑁 ⊆ ∪ 𝐽) → ((int‘𝐽)‘(𝑀 ∩ 𝑁)) = (((int‘𝐽)‘𝑀) ∩ ((int‘𝐽)‘𝑁)))
27	23, 24, 25, 26	syl3anc 1374	. . 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 3973	. 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 17	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝐴 ∩ 𝐵) ⊆ ∪ 𝐽)
31		ssinss1 4200	. . . . 5 ⊢ (𝑀 ⊆ ∪ 𝐽 → (𝑀 ∩ 𝑁) ⊆ ∪ 𝐽)
32	5, 31	syl 17	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → (𝑀 ∩ 𝑁) ⊆ ∪ 𝐽)
33	3	neiint 23060	. . . 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 1374	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝐴)) → ((𝑀 ∩ 𝑁) ∈ ((nei‘𝐽)‘(𝐴 ∩ 𝐵)) ↔ (𝐴 ∩ 𝐵) ⊆ ((int‘𝐽)‘(𝑀 ∩ 𝑁))))
35	34	3adant3 1133	. 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 1087   = wceq 1542   ∈ wcel 2114   ∩ cin 3902   ⊆ wss 3903  ∪ cuni 4865  ‘cfv 6500  Topctop 22849  intcnt 22973  neicnei 23053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-top 22850  df-cld 22975  df-ntr 22976  df-cls 22977  df-nei 23054

This theorem is referenced by: (None)