Theorem neiss 20723

Description: Any neighborhood of a set 𝑆 is also a neighborhood of any subset 𝑅 ⊆ 𝑆. Theorem 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 2610	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	neii1 20720	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁 ⊆ ∪ 𝐽)
3	2	3adant3 1074	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → 𝑁 ⊆ ∪ 𝐽)
4		neii2 20722	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))
5	4	3adant3 1074	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))
6		sstr2 3575	. . . . . 6 ⊢ (𝑅 ⊆ 𝑆 → (𝑆 ⊆ 𝑔 → 𝑅 ⊆ 𝑔))
7	6	anim1d 586	. . . . 5 ⊢ (𝑅 ⊆ 𝑆 → ((𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁) → (𝑅 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)))
8	7	reximdv 2999	. . . 4 ⊢ (𝑅 ⊆ 𝑆 → (∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁) → ∃𝑔 ∈ 𝐽 (𝑅 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)))
9	8	3ad2ant3 1077	. . 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 15	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → ∃𝑔 ∈ 𝐽 (𝑅 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))
11		simp1 1054	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → 𝐽 ∈ Top)
12		simp3 1056	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → 𝑅 ⊆ 𝑆)
13	1	neiss2 20715	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ ∪ 𝐽)
14	13	3adant3 1074	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → 𝑆 ⊆ ∪ 𝐽)
15	12, 14	sstrd 3578	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → 𝑅 ⊆ ∪ 𝐽)
16	1	isnei 20717	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑅 ⊆ ∪ 𝐽) → (𝑁 ∈ ((nei‘𝐽)‘𝑅) ↔ (𝑁 ⊆ ∪ 𝐽 ∧ ∃𝑔 ∈ 𝐽 (𝑅 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
17	11, 15, 16	syl2anc 691	. 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 959	1 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → 𝑁 ∈ ((nei‘𝐽)‘𝑅))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977  ∃wrex 2897   ⊆ wss 3540  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  neicnei 20711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-top 20521  df-nei 20712

This theorem is referenced by:  neips  20727  neissex  20741