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Theorem opnneissb 21406
Description: An open set is a neighborhood of any of its subsets. (Contributed by FL, 2-Oct-2006.)
Hypothesis
Ref Expression
neips.1 𝑋 = 𝐽
Assertion
Ref Expression
opnneissb ((𝐽 ∈ Top ∧ 𝑁𝐽𝑆𝑋) → (𝑆𝑁𝑁 ∈ ((nei‘𝐽)‘𝑆)))

Proof of Theorem opnneissb
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 neips.1 . . . . . . 7 𝑋 = 𝐽
21eltopss 21199 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁𝐽) → 𝑁𝑋)
32adantr 481 . . . . 5 (((𝐽 ∈ Top ∧ 𝑁𝐽) ∧ (𝑆𝑋𝑆𝑁)) → 𝑁𝑋)
4 ssid 3910 . . . . . . 7 𝑁𝑁
5 sseq2 3914 . . . . . . . . 9 (𝑔 = 𝑁 → (𝑆𝑔𝑆𝑁))
6 sseq1 3913 . . . . . . . . 9 (𝑔 = 𝑁 → (𝑔𝑁𝑁𝑁))
75, 6anbi12d 630 . . . . . . . 8 (𝑔 = 𝑁 → ((𝑆𝑔𝑔𝑁) ↔ (𝑆𝑁𝑁𝑁)))
87rspcev 3559 . . . . . . 7 ((𝑁𝐽 ∧ (𝑆𝑁𝑁𝑁)) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑁))
94, 8mpanr2 700 . . . . . 6 ((𝑁𝐽𝑆𝑁) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑁))
109ad2ant2l 742 . . . . 5 (((𝐽 ∈ Top ∧ 𝑁𝐽) ∧ (𝑆𝑋𝑆𝑁)) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑁))
111isnei 21395 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁𝑋 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔𝑁))))
1211ad2ant2r 743 . . . . 5 (((𝐽 ∈ Top ∧ 𝑁𝐽) ∧ (𝑆𝑋𝑆𝑁)) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁𝑋 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔𝑁))))
133, 10, 12mpbir2and 709 . . . 4 (((𝐽 ∈ Top ∧ 𝑁𝐽) ∧ (𝑆𝑋𝑆𝑁)) → 𝑁 ∈ ((nei‘𝐽)‘𝑆))
1413exp43 437 . . 3 (𝐽 ∈ Top → (𝑁𝐽 → (𝑆𝑋 → (𝑆𝑁𝑁 ∈ ((nei‘𝐽)‘𝑆)))))
15143imp 1104 . 2 ((𝐽 ∈ Top ∧ 𝑁𝐽𝑆𝑋) → (𝑆𝑁𝑁 ∈ ((nei‘𝐽)‘𝑆)))
16 ssnei 21402 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆𝑁)
1716ex 413 . . 3 (𝐽 ∈ Top → (𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝑆𝑁))
18173ad2ant1 1126 . 2 ((𝐽 ∈ Top ∧ 𝑁𝐽𝑆𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝑆𝑁))
1915, 18impbid 213 1 ((𝐽 ∈ Top ∧ 𝑁𝐽𝑆𝑋) → (𝑆𝑁𝑁 ∈ ((nei‘𝐽)‘𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wrex 3106  wss 3859   cuni 4745  cfv 6225  Topctop 21185  neicnei 21389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-top 21186  df-nei 21390
This theorem is referenced by:  opnneiss  21410
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