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Theorem neiss 23132
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 2734 . . . 4 𝐽 = 𝐽
21neii1 23129 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑁 𝐽)
323adant3 1131 . 2 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅𝑆) → 𝑁 𝐽)
4 neii2 23131 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑁))
543adant3 1131 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅𝑆) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑁))
6 sstr2 4001 . . . . . 6 (𝑅𝑆 → (𝑆𝑔𝑅𝑔))
76anim1d 611 . . . . 5 (𝑅𝑆 → ((𝑆𝑔𝑔𝑁) → (𝑅𝑔𝑔𝑁)))
87reximdv 3167 . . . 4 (𝑅𝑆 → (∃𝑔𝐽 (𝑆𝑔𝑔𝑁) → ∃𝑔𝐽 (𝑅𝑔𝑔𝑁)))
983ad2ant3 1134 . . 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 1135 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅𝑆) → 𝐽 ∈ Top)
12 simp3 1137 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅𝑆) → 𝑅𝑆)
131neiss2 23124 . . . . 5 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 𝐽)
14133adant3 1131 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅𝑆) → 𝑆 𝐽)
1512, 14sstrd 4005 . . 3 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝑅𝑆) → 𝑅 𝐽)
161isnei 23126 . . 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 1086  wcel 2105  wrex 3067  wss 3962   cuni 4911  cfv 6562  Topctop 22914  neicnei 23120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-top 22915  df-nei 23121
This theorem is referenced by:  neips  23136  neissex  23150
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