Theorem neiss 14692

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 2206	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	neii1 14689	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁 ⊆ ∪ 𝐽)
3	2	3adant3 1020	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → 𝑁 ⊆ ∪ 𝐽)
4		neii2 14691	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))
5	4	3adant3 1020	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → ∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))
6		sstr2 3204	. . . . . 6 ⊢ (𝑅 ⊆ 𝑆 → (𝑆 ⊆ 𝑔 → 𝑅 ⊆ 𝑔))
7	6	anim1d 336	. . . . 5 ⊢ (𝑅 ⊆ 𝑆 → ((𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁) → (𝑅 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)))
8	7	reximdv 2608	. . . 4 ⊢ (𝑅 ⊆ 𝑆 → (∃𝑔 ∈ 𝐽 (𝑆 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁) → ∃𝑔 ∈ 𝐽 (𝑅 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁)))
9	8	3ad2ant3 1023	. . 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 1000	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → 𝐽 ∈ Top)
12		simp3 1002	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → 𝑅 ⊆ 𝑆)
13	1	neiss2 14684	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ ∪ 𝐽)
14	13	3adant3 1020	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → 𝑆 ⊆ ∪ 𝐽)
15	12, 14	sstrd 3207	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → 𝑅 ⊆ ∪ 𝐽)
16	1	isnei 14686	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑅 ⊆ ∪ 𝐽) → (𝑁 ∈ ((nei‘𝐽)‘𝑅) ↔ (𝑁 ⊆ ∪ 𝐽 ∧ ∃𝑔 ∈ 𝐽 (𝑅 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑁))))
17	11, 15, 16	syl2anc 411	. 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 947	1 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅 ⊆ 𝑆) → 𝑁 ∈ ((nei‘𝐽)‘𝑅))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 981   ∈ wcel 2177  ∃wrex 2486   ⊆ wss 3170  ∪ cuni 3855  ‘cfv 5279  Topctop 14539  neicnei 14680

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-coll 4166  ax-sep 4169  ax-pow 4225  ax-pr 4260

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-iun 3934  df-br 4051  df-opab 4113  df-mpt 4114  df-id 4347  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-top 14540  df-nei 14681

This theorem is referenced by:  neipsm  14696  neissex  14707