MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  flimss1 Structured version   Visualization version   GIF version

Theorem flimss1 23406
Description: A limit point of a filter is a limit point in a coarser topology. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.)
Assertion
Ref Expression
flimss1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) → (𝐾 fLim 𝐹) ⊆ (𝐽 fLim 𝐹))

Proof of Theorem flimss1
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2731 . . . . . . 7 𝐾 = 𝐾
21flimelbas 23401 . . . . . 6 (𝑥 ∈ (𝐾 fLim 𝐹) → 𝑥 𝐾)
32adantl 482 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝑥 𝐾)
4 simpl2 1192 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐹 ∈ (Fil‘𝑋))
5 filunibas 23314 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
64, 5syl 17 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐹 = 𝑋)
71flimfil 23402 . . . . . . . 8 (𝑥 ∈ (𝐾 fLim 𝐹) → 𝐹 ∈ (Fil‘ 𝐾))
87adantl 482 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐹 ∈ (Fil‘ 𝐾))
9 filunibas 23314 . . . . . . 7 (𝐹 ∈ (Fil‘ 𝐾) → 𝐹 = 𝐾)
108, 9syl 17 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐹 = 𝐾)
116, 10eqtr3d 2773 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝑋 = 𝐾)
123, 11eleqtrrd 2835 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝑥𝑋)
13 simpl1 1191 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐽 ∈ (TopOn‘𝑋))
14 topontop 22344 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
1513, 14syl 17 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐽 ∈ Top)
16 flimtop 23398 . . . . . . 7 (𝑥 ∈ (𝐾 fLim 𝐹) → 𝐾 ∈ Top)
1716adantl 482 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐾 ∈ Top)
18 toponuni 22345 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1913, 18syl 17 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝑋 = 𝐽)
2019, 11eqtr3d 2773 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐽 = 𝐾)
21 simpl3 1193 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐽𝐾)
22 eqid 2731 . . . . . . 7 𝐽 = 𝐽
2322, 1topssnei 22557 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐽 = 𝐾) ∧ 𝐽𝐾) → ((nei‘𝐽)‘{𝑥}) ⊆ ((nei‘𝐾)‘{𝑥}))
2415, 17, 20, 21, 23syl31anc 1373 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → ((nei‘𝐽)‘{𝑥}) ⊆ ((nei‘𝐾)‘{𝑥}))
25 flimneiss 23399 . . . . . 6 (𝑥 ∈ (𝐾 fLim 𝐹) → ((nei‘𝐾)‘{𝑥}) ⊆ 𝐹)
2625adantl 482 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → ((nei‘𝐾)‘{𝑥}) ⊆ 𝐹)
2724, 26sstrd 3988 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹)
28 elflim 23404 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥𝑋 ∧ ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹)))
2913, 4, 28syl2anc 584 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥𝑋 ∧ ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹)))
3012, 27, 29mpbir2and 711 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝑥 ∈ (𝐽 fLim 𝐹))
3130ex 413 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) → (𝑥 ∈ (𝐾 fLim 𝐹) → 𝑥 ∈ (𝐽 fLim 𝐹)))
3231ssrdv 3984 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽𝐾) → (𝐾 fLim 𝐹) ⊆ (𝐽 fLim 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wss 3944  {csn 4622   cuni 4901  cfv 6532  (class class class)co 7393  Topctop 22324  TopOnctopon 22341  neicnei 22530  Filcfil 23278   fLim cflim 23367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-fbas 20875  df-top 22325  df-topon 22342  df-ntr 22453  df-nei 22531  df-fil 23279  df-flim 23372
This theorem is referenced by:  flimcf  23415
  Copyright terms: Public domain W3C validator