Theorem innei 13214

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 2175	. . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	neii1 13198	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁 ⊆ ∪ 𝐽)
3		ssinss1 3362	. . . 4 ⊢ (𝑁 ⊆ ∪ 𝐽 → (𝑁 ∩ 𝑀) ⊆ ∪ 𝐽)
4	2, 3	syl 14	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁 ∩ 𝑀) ⊆ ∪ 𝐽)
5	4	3adant3 1017	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁 ∩ 𝑀) ⊆ ∪ 𝐽)
6		neii2 13200	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃ℎ ∈ 𝐽 (𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁))
7		neii2 13200	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑣 ∈ 𝐽 (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑀))
8	6, 7	anim12dan 600	. . . 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 13052	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ ℎ ∈ 𝐽 ∧ 𝑣 ∈ 𝐽) → (ℎ ∩ 𝑣) ∈ 𝐽)
10	9	3expa 1203	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ ℎ ∈ 𝐽) ∧ 𝑣 ∈ 𝐽) → (ℎ ∩ 𝑣) ∈ 𝐽)
11		ssin 3355	. . . . . . . . . . . . 13 ⊢ ((𝑆 ⊆ ℎ ∧ 𝑆 ⊆ 𝑣) ↔ 𝑆 ⊆ (ℎ ∩ 𝑣))
12	11	biimpi 120	. . . . . . . . . . . 12 ⊢ ((𝑆 ⊆ ℎ ∧ 𝑆 ⊆ 𝑣) → 𝑆 ⊆ (ℎ ∩ 𝑣))
13		ss2in 3361	. . . . . . . . . . . 12 ⊢ ((ℎ ⊆ 𝑁 ∧ 𝑣 ⊆ 𝑀) → (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀))
14	12, 13	anim12i 338	. . . . . . . . . . 11 ⊢ (((𝑆 ⊆ ℎ ∧ 𝑆 ⊆ 𝑣) ∧ (ℎ ⊆ 𝑁 ∧ 𝑣 ⊆ 𝑀)) → (𝑆 ⊆ (ℎ ∩ 𝑣) ∧ (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀)))
15	14	an4s 588	. . . . . . . . . 10 ⊢ (((𝑆 ⊆ ℎ ∧ ℎ ⊆ 𝑁) ∧ (𝑆 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑀)) → (𝑆 ⊆ (ℎ ∩ 𝑣) ∧ (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀)))
16		sseq2 3177	. . . . . . . . . . . 12 ⊢ (𝑔 = (ℎ ∩ 𝑣) → (𝑆 ⊆ 𝑔 ↔ 𝑆 ⊆ (ℎ ∩ 𝑣)))
17		sseq1 3176	. . . . . . . . . . . 12 ⊢ (𝑔 = (ℎ ∩ 𝑣) → (𝑔 ⊆ (𝑁 ∩ 𝑀) ↔ (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀)))
18	16, 17	anbi12d 473	. . . . . . . . . . 11 ⊢ (𝑔 = (ℎ ∩ 𝑣) → ((𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)) ↔ (𝑆 ⊆ (ℎ ∩ 𝑣) ∧ (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀))))
19	18	rspcev 2839	. . . . . . . . . 10 ⊢ (((ℎ ∩ 𝑣) ∈ 𝐽 ∧ (𝑆 ⊆ (ℎ ∩ 𝑣) ∧ (ℎ ∩ 𝑣) ⊆ (𝑁 ∩ 𝑀))) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))
20	10, 15, 19	syl2an 289	. . . . . . . . 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 375	. . . . . . . 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 568	. . . . . . 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 2592	. . . . . 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 2595	. . . . 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 257	. . . 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 282	. . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆))) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))
27	26	3impb 1199	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))
28	1	neiss2 13193	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ ∪ 𝐽)
29	1	isnei 13195	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((𝑁 ∩ 𝑀) ∈ ((nei‘𝐽)‘𝑆) ↔ ((𝑁 ∩ 𝑀) ⊆ ∪ 𝐽 ∧ ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ (𝑁 ∩ 𝑀)))))
30	28, 29	syldan 282	. . 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 1017	. 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 944	1 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑀 ∈ ((nei‘𝐽)‘𝑆)) → (𝑁 ∩ 𝑀) ∈ ((nei‘𝐽)‘𝑆))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2146  ∃wrex 2454   ∩ cin 3126   ⊆ wss 3127  ∪ cuni 3805  ‘cfv 5208  Topctop 13046  neicnei 13189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-top 13047  df-nei 13190

This theorem is referenced by: (None)