Theorem neiss 12790

Description: Any neighborhood of a set 𝑆 is also a neighborhood of any subset 𝑅 ⊆ 𝑆. Similar to Proposition 1 of [BourbakiTop1] p. I.2. (Contributed by FL, 25-Sep-2006.)

Assertion

Ref	Expression
neiss	⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → 𝑁 ∈ ((nei‘𝐽)‘𝑅))

Proof of Theorem neiss

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2165	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	neii1 12787	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁 ⊆ ∪ 𝐽)
3	2	3adant3 1007	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → 𝑁 ⊆ ∪ 𝐽)
4		neii2 12789	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))
5	4	3adant3 1007	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))
6		sstr2 3149	. . . . . 6 ⊢ (𝑅 ⊆ 𝑆 → (𝑆 ⊆ 𝑔 → 𝑅 ⊆ 𝑔))
7	6	anim1d 334	. . . . 5 ⊢ (𝑅 ⊆ 𝑆 → ((𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁) → (𝑅 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)))
8	7	reximdv 2567	. . . 4 ⊢ (𝑅 ⊆ 𝑆 → (∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁) → ∃𝑔 ∈ 𝐽 (𝑅 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)))
9	8	3ad2ant3 1010	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → (∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁) → ∃𝑔 ∈ 𝐽 (𝑅 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)))
10	5, 9	mpd 13	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → ∃𝑔 ∈ 𝐽 (𝑅 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))
11		simp1 987	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → 𝐽 ∈ Top)
12		simp3 989	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → 𝑅 ⊆ 𝑆)
13	1	neiss2 12782	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ ∪ 𝐽)
14	13	3adant3 1007	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → 𝑆 ⊆ ∪ 𝐽)
15	12, 14	sstrd 3152	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → 𝑅 ⊆ ∪ 𝐽)
16	1	isnei 12784	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑅 ⊆ ∪ 𝐽) → (𝑁 ∈ ((nei‘𝐽)‘𝑅) ↔ (𝑁 ⊆ ∪ 𝐽 ∧ ∃𝑔 ∈ 𝐽 (𝑅 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
17	11, 15, 16	syl2anc 409	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → (𝑁 ∈ ((nei‘𝐽)‘𝑅) ↔ (𝑁 ⊆ ∪ 𝐽 ∧ ∃𝑔 ∈ 𝐽 (𝑅 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
18	3, 10, 17	mpbir2and 934	1 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → 𝑁 ∈ ((nei‘𝐽)‘𝑅))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 968   ∈ wcel 2136  ∃wrex 2445   ⊆ wss 3116  ∪ cuni 3789  ‘cfv 5188  Topctop 12635  neicnei 12778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-top 12636  df-nei 12779

This theorem is referenced by:  neipsm  12794  neissex  12805