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Theorem neiss 23117
Description: Any neighborhood of a set 𝑆 is also a neighborhood of any subset 𝑅𝑆. 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) ∧ 𝑅𝑆) → 𝑁 ∈ ((nei‘𝐽)‘𝑅))

Proof of Theorem neiss
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . 4 𝐽 = 𝐽
21neii1 23114 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁 𝐽)
323adant3 1133 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅𝑆) → 𝑁 𝐽)
4 neii2 23116 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑁))
543adant3 1133 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅𝑆) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑁))
6 sstr2 3990 . . . . . 6 (𝑅𝑆 → (𝑆𝑔𝑅𝑔))
76anim1d 611 . . . . 5 (𝑅𝑆 → ((𝑆𝑔𝑔𝑁) → (𝑅𝑔𝑔𝑁)))
87reximdv 3170 . . . 4 (𝑅𝑆 → (∃𝑔𝐽 (𝑆𝑔𝑔𝑁) → ∃𝑔𝐽 (𝑅𝑔𝑔𝑁)))
983ad2ant3 1136 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅𝑆) → (∃𝑔𝐽 (𝑆𝑔𝑔𝑁) → ∃𝑔𝐽 (𝑅𝑔𝑔𝑁)))
105, 9mpd 15 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅𝑆) → ∃𝑔𝐽 (𝑅𝑔𝑔𝑁))
11 simp1 1137 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅𝑆) → 𝐽 ∈ Top)
12 simp3 1139 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅𝑆) → 𝑅𝑆)
131neiss2 23109 . . . . 5 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 𝐽)
14133adant3 1133 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅𝑆) → 𝑆 𝐽)
1512, 14sstrd 3994 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅𝑆) → 𝑅 𝐽)
161isnei 23111 . . 3 ((𝐽 ∈ Top ∧ 𝑅 𝐽) → (𝑁 ∈ ((nei‘𝐽)‘𝑅) ↔ (𝑁 𝐽 ∧ ∃𝑔𝐽 (𝑅𝑔𝑔𝑁))))
1711, 15, 16syl2anc 584 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅𝑆) → (𝑁 ∈ ((nei‘𝐽)‘𝑅) ↔ (𝑁 𝐽 ∧ ∃𝑔𝐽 (𝑅𝑔𝑔𝑁))))
183, 10, 17mpbir2and 713 1 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅𝑆) → 𝑁 ∈ ((nei‘𝐽)‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087  wcel 2108  wrex 3070  wss 3951   cuni 4907  cfv 6561  Topctop 22899  neicnei 23105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-top 22900  df-nei 23106
This theorem is referenced by:  neips  23121  neissex  23135
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