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Theorem flimss1 23477
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 2733 . . . . . . 7 βˆͺ 𝐾 = βˆͺ 𝐾
21flimelbas 23472 . . . . . 6 (π‘₯ ∈ (𝐾 fLim 𝐹) β†’ π‘₯ ∈ βˆͺ 𝐾)
32adantl 483 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ π‘₯ ∈ βˆͺ 𝐾)
4 simpl2 1193 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
5 filunibas 23385 . . . . . . 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 23473 . . . . . . . 8 (π‘₯ ∈ (𝐾 fLim 𝐹) β†’ 𝐹 ∈ (Filβ€˜βˆͺ 𝐾))
87adantl 483 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ 𝐹 ∈ (Filβ€˜βˆͺ 𝐾))
9 filunibas 23385 . . . . . . 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 2775 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ 𝑋 = βˆͺ 𝐾)
123, 11eleqtrrd 2837 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ π‘₯ ∈ 𝑋)
13 simpl1 1192 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
14 topontop 22415 . . . . . . 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 23469 . . . . . . 7 (π‘₯ ∈ (𝐾 fLim 𝐹) β†’ 𝐾 ∈ Top)
1716adantl 483 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ 𝐾 ∈ Top)
18 toponuni 22416 . . . . . . . 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 2775 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ βˆͺ 𝐽 = βˆͺ 𝐾)
21 simpl3 1194 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ 𝐽 βŠ† 𝐾)
22 eqid 2733 . . . . . . 7 βˆͺ 𝐽 = βˆͺ 𝐽
2322, 1topssnei 22628 . . . . . 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 1374 . . . . 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 23470 . . . . . 6 (π‘₯ ∈ (𝐾 fLim 𝐹) β†’ ((neiβ€˜πΎ)β€˜{π‘₯}) βŠ† 𝐹)
2625adantl 483 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ ((neiβ€˜πΎ)β€˜{π‘₯}) βŠ† 𝐹)
2724, 26sstrd 3993 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ ((neiβ€˜π½)β€˜{π‘₯}) βŠ† 𝐹)
28 elflim 23475 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (π‘₯ ∈ (𝐽 fLim 𝐹) ↔ (π‘₯ ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{π‘₯}) βŠ† 𝐹)))
2913, 4, 28syl2anc 585 . . . 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 712 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ (𝐾 fLim 𝐹)) β†’ π‘₯ ∈ (𝐽 fLim 𝐹))
3130ex 414 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹) ∧ 𝐽 βŠ† 𝐾) β†’ (π‘₯ ∈ (𝐾 fLim 𝐹) β†’ π‘₯ ∈ (𝐽 fLim 𝐹)))
3231ssrdv 3989 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   βŠ† wss 3949  {csn 4629  βˆͺ cuni 4909  β€˜cfv 6544  (class class class)co 7409  Topctop 22395  TopOnctopon 22412  neicnei 22601  Filcfil 23349   fLim cflim 23438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-fbas 20941  df-top 22396  df-topon 22413  df-ntr 22524  df-nei 22602  df-fil 23350  df-flim 23443
This theorem is referenced by:  flimcf  23486
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