MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opnneissb Structured version   Visualization version   GIF version

Theorem opnneissb 20966
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 20760 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑁𝐽) → 𝑁𝑋)
32adantr 480 . . . . 5 (((𝐽 ∈ Top ∧ 𝑁𝐽) ∧ (𝑆𝑋𝑆𝑁)) → 𝑁𝑋)
4 ssid 3657 . . . . . . 7 𝑁𝑁
5 sseq2 3660 . . . . . . . . 9 (𝑔 = 𝑁 → (𝑆𝑔𝑆𝑁))
6 sseq1 3659 . . . . . . . . 9 (𝑔 = 𝑁 → (𝑔𝑁𝑁𝑁))
75, 6anbi12d 747 . . . . . . . 8 (𝑔 = 𝑁 → ((𝑆𝑔𝑔𝑁) ↔ (𝑆𝑁𝑁𝑁)))
87rspcev 3340 . . . . . . 7 ((𝑁𝐽 ∧ (𝑆𝑁𝑁𝑁)) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑁))
94, 8mpanr2 720 . . . . . 6 ((𝑁𝐽𝑆𝑁) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑁))
109ad2ant2l 797 . . . . 5 (((𝐽 ∈ Top ∧ 𝑁𝐽) ∧ (𝑆𝑋𝑆𝑁)) → ∃𝑔𝐽 (𝑆𝑔𝑔𝑁))
111isnei 20955 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁𝑋 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔𝑁))))
1211ad2ant2r 798 . . . . 5 (((𝐽 ∈ Top ∧ 𝑁𝐽) ∧ (𝑆𝑋𝑆𝑁)) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) ↔ (𝑁𝑋 ∧ ∃𝑔𝐽 (𝑆𝑔𝑔𝑁))))
133, 10, 12mpbir2and 977 . . . 4 (((𝐽 ∈ Top ∧ 𝑁𝐽) ∧ (𝑆𝑋𝑆𝑁)) → 𝑁 ∈ ((nei‘𝐽)‘𝑆))
1413exp43 639 . . 3 (𝐽 ∈ Top → (𝑁𝐽 → (𝑆𝑋 → (𝑆𝑁𝑁 ∈ ((nei‘𝐽)‘𝑆)))))
15143imp 1275 . 2 ((𝐽 ∈ Top ∧ 𝑁𝐽𝑆𝑋) → (𝑆𝑁𝑁 ∈ ((nei‘𝐽)‘𝑆)))
16 ssnei 20962 . . . 4 ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆𝑁)
1716ex 449 . . 3 (𝐽 ∈ Top → (𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝑆𝑁))
18173ad2ant1 1102 . 2 ((𝐽 ∈ Top ∧ 𝑁𝐽𝑆𝑋) → (𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝑆𝑁))
1915, 18impbid 202 1 ((𝐽 ∈ Top ∧ 𝑁𝐽𝑆𝑋) → (𝑆𝑁𝑁 ∈ ((nei‘𝐽)‘𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wrex 2942  wss 3607   cuni 4468  cfv 5926  Topctop 20746  neicnei 20949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-top 20747  df-nei 20950
This theorem is referenced by:  opnneiss  20970
  Copyright terms: Public domain W3C validator