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

Theorem opnnei 20864
Description: A set is open iff it is a neighborhood of all of its points. (Contributed by Jeff Hankins, 15-Sep-2009.)
Assertion
Ref Expression
opnnei (𝐽 ∈ Top → (𝑆𝐽 ↔ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
Distinct variable groups:   𝑥,𝐽   𝑥,𝑆

Proof of Theorem opnnei
StepHypRef Expression
1 0opn 20649 . . . . 5 (𝐽 ∈ Top → ∅ ∈ 𝐽)
21adantr 481 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 = ∅) → ∅ ∈ 𝐽)
3 eleq1 2686 . . . . 5 (𝑆 = ∅ → (𝑆𝐽 ↔ ∅ ∈ 𝐽))
43adantl 482 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 = ∅) → (𝑆𝐽 ↔ ∅ ∈ 𝐽))
52, 4mpbird 247 . . 3 ((𝐽 ∈ Top ∧ 𝑆 = ∅) → 𝑆𝐽)
6 rzal 4051 . . . 4 (𝑆 = ∅ → ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
76adantl 482 . . 3 ((𝐽 ∈ Top ∧ 𝑆 = ∅) → ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
85, 72thd 255 . 2 ((𝐽 ∈ Top ∧ 𝑆 = ∅) → (𝑆𝐽 ↔ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
9 opnneip 20863 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆𝐽𝑥𝑆) → 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
1093expia 1264 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝐽) → (𝑥𝑆𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
1110ralrimiv 2961 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝐽) → ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
1211ex 450 . . . 4 (𝐽 ∈ Top → (𝑆𝐽 → ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
1312adantr 481 . . 3 ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (𝑆𝐽 → ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
14 df-ne 2791 . . . . . 6 (𝑆 ≠ ∅ ↔ ¬ 𝑆 = ∅)
15 r19.2z 4038 . . . . . . 7 ((𝑆 ≠ ∅ ∧ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
1615ex 450 . . . . . 6 (𝑆 ≠ ∅ → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → ∃𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
1714, 16sylbir 225 . . . . 5 𝑆 = ∅ → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → ∃𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
18 eqid 2621 . . . . . . . 8 𝐽 = 𝐽
1918neii1 20850 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑆 𝐽)
2019ex 450 . . . . . 6 (𝐽 ∈ Top → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 𝐽))
2120rexlimdvw 3029 . . . . 5 (𝐽 ∈ Top → (∃𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 𝐽))
2217, 21sylan9r 689 . . . 4 ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 𝐽))
2318ntrss2 20801 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
2423adantr 481 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ ∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
25 vex 3193 . . . . . . . . . . . . 13 𝑥 ∈ V
2625snss 4293 . . . . . . . . . . . 12 (𝑥 ∈ ((int‘𝐽)‘𝑆) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆))
2726ralbii 2976 . . . . . . . . . . 11 (∀𝑥𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆) ↔ ∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆))
28 dfss3 3578 . . . . . . . . . . . . 13 (𝑆 ⊆ ((int‘𝐽)‘𝑆) ↔ ∀𝑥𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆))
2928biimpri 218 . . . . . . . . . . . 12 (∀𝑥𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
3029adantl 482 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ ∀𝑥𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
3127, 30sylan2br 493 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ ∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
3224, 31eqssd 3605 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ ∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → ((int‘𝐽)‘𝑆) = 𝑆)
3332ex 450 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆) → ((int‘𝐽)‘𝑆) = 𝑆))
3425snss 4293 . . . . . . . . . . . 12 (𝑥𝑆 ↔ {𝑥} ⊆ 𝑆)
35 sstr2 3595 . . . . . . . . . . . . . 14 ({𝑥} ⊆ 𝑆 → (𝑆 𝐽 → {𝑥} ⊆ 𝐽))
3635com12 32 . . . . . . . . . . . . 13 (𝑆 𝐽 → ({𝑥} ⊆ 𝑆 → {𝑥} ⊆ 𝐽))
3736adantl 482 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ({𝑥} ⊆ 𝑆 → {𝑥} ⊆ 𝐽))
3834, 37syl5bi 232 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (𝑥𝑆 → {𝑥} ⊆ 𝐽))
3938imp 445 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ 𝑥𝑆) → {𝑥} ⊆ 𝐽)
4018neiint 20848 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ {𝑥} ⊆ 𝐽𝑆 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
41403com23 1268 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑆 𝐽 ∧ {𝑥} ⊆ 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
42413expa 1262 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ {𝑥} ⊆ 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
4339, 42syldan 487 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ 𝑥𝑆) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
4443ralbidva 2981 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ ∀𝑥𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
4518isopn3 20810 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (𝑆𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆))
4633, 44, 453imtr4d 283 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆𝐽))
4746ex 450 . . . . . 6 (𝐽 ∈ Top → (𝑆 𝐽 → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆𝐽)))
4847com23 86 . . . . 5 (𝐽 ∈ Top → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑆 𝐽𝑆𝐽)))
4948adantr 481 . . . 4 ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑆 𝐽𝑆𝐽)))
5022, 49mpdd 43 . . 3 ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆𝐽))
5113, 50impbid 202 . 2 ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (𝑆𝐽 ↔ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
528, 51pm2.61dan 831 1 (𝐽 ∈ Top → (𝑆𝐽 ↔ ∀𝑥𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2908  wrex 2909  wss 3560  c0 3897  {csn 4155   cuni 4409  cfv 5857  Topctop 20638  intcnt 20761  neicnei 20841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-top 20639  df-ntr 20764  df-nei 20842
This theorem is referenced by:  neiptopreu  20877  flimcf  21726
  Copyright terms: Public domain W3C validator