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Theorem flimss1 23484
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 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) β†’ (𝐾 fLim 𝐹) βŠ† (𝐽 fLim 𝐹))
Proof of Theorem flimss1
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . . . . . 7 βˆͺ 𝐾 = βˆͺ 𝐾
21flimelbas 23479 . . . . . 6 (π‘₯ ∈ (𝐾 fLim 𝐹) β†’ π‘₯ ∈ βˆͺ 𝐾)
32adantl 482 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ π‘₯ ∈ βˆͺ 𝐾)
4 simpl2 1192 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
5 filunibas 23392 . . . . . . 7 (𝐹 ∈ (Filβ€˜π‘‹) β†’ βˆͺ 𝐹 = 𝑋)
64, 5syl 17 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ βˆͺ 𝐹 = 𝑋)
71flimfil 23480 . . . . . . . 8 (π‘₯ ∈ (𝐾 fLim 𝐹) β†’ 𝐹 ∈ (Filβ€˜βˆͺ 𝐾))
87adantl 482 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ 𝐹 ∈ (Filβ€˜βˆͺ 𝐾))
9 filunibas 23392 . . . . . . 7 (𝐹 ∈ (Filβ€˜βˆͺ 𝐾) β†’ βˆͺ 𝐹 = βˆͺ 𝐾)
108, 9syl 17 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ βˆͺ 𝐹 = βˆͺ 𝐾)
116, 10eqtr3d 2774 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ 𝑋 = βˆͺ 𝐾)
123, 11eleqtrrd 2836 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ π‘₯ ∈ 𝑋)
13 simpl1 1191 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
14 topontop 22422 . . . . . . 7 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
1513, 14syl 17 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ 𝐽 ∈ Top)
16 flimtop 23476 . . . . . . 7 (π‘₯ ∈ (𝐾 fLim 𝐹) β†’ 𝐾 ∈ Top)
1716adantl 482 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ 𝐾 ∈ Top)
18 toponuni 22423 . . . . . . . 8 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
1913, 18syl 17 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ 𝑋 = βˆͺ 𝐽)
2019, 11eqtr3d 2774 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ βˆͺ 𝐽 = βˆͺ 𝐾)
21 simpl3 1193 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ 𝐽 βŠ† 𝐾)
22 eqid 2732 . . . . . . 7 βˆͺ 𝐽 = βˆͺ 𝐽
2322, 1topssnei 22635 . . . . . 6 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ βˆͺ 𝐽 = βˆͺ 𝐾) ∧ 𝐽 βŠ† 𝐾) β†’ ((neiβ€˜π½)β€˜{π‘₯}) βŠ† ((neiβ€˜πΎ)β€˜{π‘₯}))
2415, 17, 20, 21, 23syl31anc 1373 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ ((neiβ€˜π½)β€˜{π‘₯}) βŠ† ((neiβ€˜πΎ)β€˜{π‘₯}))
25 flimneiss 23477 . . . . . 6 (π‘₯ ∈ (𝐾 fLim 𝐹) β†’ ((neiβ€˜πΎ)β€˜{π‘₯}) βŠ† 𝐹)
2625adantl 482 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ ((neiβ€˜πΎ)β€˜{π‘₯}) βŠ† 𝐹)
2724, 26sstrd 3992 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ ((neiβ€˜π½)β€˜{π‘₯}) βŠ† 𝐹)
28 elflim 23482 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (π‘₯ ∈ (𝐽 fLim 𝐹) ↔ (π‘₯ ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{π‘₯}) βŠ† 𝐹)))
2913, 4, 28syl2anc 584 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ (π‘₯ ∈ (𝐽 fLim 𝐹) ↔ (π‘₯ ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{π‘₯}) βŠ† 𝐹)))
3012, 27, 29mpbir2and 711 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ π‘₯ ∈ (𝐽 fLim 𝐹))
3130ex 413 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) β†’ (π‘₯ ∈ (𝐾 fLim 𝐹) β†’ π‘₯ ∈ (𝐽 fLim 𝐹)))
3231ssrdv 3988 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) β†’ (𝐾 fLim 𝐹) βŠ† (𝐽 fLim 𝐹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   βŠ† wss 3948  {csn 4628  βˆͺ cuni 4908  β€˜cfv 6543  (class class class)co 7411  Topctop 22402  TopOnctopon 22419  neicnei 22608  Filcfil 23356   fLim cflim 23445
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-fbas 20947  df-top 22403  df-topon 22420  df-ntr 22531  df-nei 22609  df-fil 23357  df-flim 23450
This theorem is referenced by:  flimcf  23493
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