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Theorem elflim2 23689
Description: The predicate "is a limit point of a filter." (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
flimval.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
elflim2 (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ βˆͺ ran Fil ∧ 𝐹 βŠ† 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)))

Proof of Theorem elflim2
Dummy variables π‘₯ 𝑓 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 anass 468 . 2 ((((𝐽 ∈ Top ∧ 𝐹 ∈ βˆͺ ran Fil) ∧ 𝐹 βŠ† 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ βˆͺ ran Fil) ∧ (𝐹 βŠ† 𝒫 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹))))
2 df-3an 1088 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ βˆͺ ran Fil ∧ 𝐹 βŠ† 𝒫 𝑋) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ βˆͺ ran Fil) ∧ 𝐹 βŠ† 𝒫 𝑋))
32anbi1i 623 . 2 (((𝐽 ∈ Top ∧ 𝐹 ∈ βˆͺ ran Fil ∧ 𝐹 βŠ† 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)) ↔ (((𝐽 ∈ Top ∧ 𝐹 ∈ βˆͺ ran Fil) ∧ 𝐹 βŠ† 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)))
4 df-flim 23664 . . . 4 fLim = (𝑗 ∈ Top, 𝑓 ∈ βˆͺ ran Fil ↦ {π‘₯ ∈ βˆͺ 𝑗 ∣ (((neiβ€˜π‘—)β€˜{π‘₯}) βŠ† 𝑓 ∧ 𝑓 βŠ† 𝒫 βˆͺ 𝑗)})
54elmpocl 7651 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ (𝐽 ∈ Top ∧ 𝐹 ∈ βˆͺ ran Fil))
6 flimval.1 . . . . . 6 𝑋 = βˆͺ 𝐽
76flimval 23688 . . . . 5 ((𝐽 ∈ Top ∧ 𝐹 ∈ βˆͺ ran Fil) β†’ (𝐽 fLim 𝐹) = {π‘₯ ∈ 𝑋 ∣ (((neiβ€˜π½)β€˜{π‘₯}) βŠ† 𝐹 ∧ 𝐹 βŠ† 𝒫 𝑋)})
87eleq2d 2818 . . . 4 ((𝐽 ∈ Top ∧ 𝐹 ∈ βˆͺ ran Fil) β†’ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ 𝐴 ∈ {π‘₯ ∈ 𝑋 ∣ (((neiβ€˜π½)β€˜{π‘₯}) βŠ† 𝐹 ∧ 𝐹 βŠ† 𝒫 𝑋)}))
9 sneq 4638 . . . . . . . . . 10 (π‘₯ = 𝐴 β†’ {π‘₯} = {𝐴})
109fveq2d 6895 . . . . . . . . 9 (π‘₯ = 𝐴 β†’ ((neiβ€˜π½)β€˜{π‘₯}) = ((neiβ€˜π½)β€˜{𝐴}))
1110sseq1d 4013 . . . . . . . 8 (π‘₯ = 𝐴 β†’ (((neiβ€˜π½)β€˜{π‘₯}) βŠ† 𝐹 ↔ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹))
1211anbi1d 629 . . . . . . 7 (π‘₯ = 𝐴 β†’ ((((neiβ€˜π½)β€˜{π‘₯}) βŠ† 𝐹 ∧ 𝐹 βŠ† 𝒫 𝑋) ↔ (((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹 ∧ 𝐹 βŠ† 𝒫 𝑋)))
1312biancomd 463 . . . . . 6 (π‘₯ = 𝐴 β†’ ((((neiβ€˜π½)β€˜{π‘₯}) βŠ† 𝐹 ∧ 𝐹 βŠ† 𝒫 𝑋) ↔ (𝐹 βŠ† 𝒫 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)))
1413elrab 3683 . . . . 5 (𝐴 ∈ {π‘₯ ∈ 𝑋 ∣ (((neiβ€˜π½)β€˜{π‘₯}) βŠ† 𝐹 ∧ 𝐹 βŠ† 𝒫 𝑋)} ↔ (𝐴 ∈ 𝑋 ∧ (𝐹 βŠ† 𝒫 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)))
15 an12 642 . . . . 5 ((𝐴 ∈ 𝑋 ∧ (𝐹 βŠ† 𝒫 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)) ↔ (𝐹 βŠ† 𝒫 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)))
1614, 15bitri 275 . . . 4 (𝐴 ∈ {π‘₯ ∈ 𝑋 ∣ (((neiβ€˜π½)β€˜{π‘₯}) βŠ† 𝐹 ∧ 𝐹 βŠ† 𝒫 𝑋)} ↔ (𝐹 βŠ† 𝒫 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)))
178, 16bitrdi 287 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ βˆͺ ran Fil) β†’ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐹 βŠ† 𝒫 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹))))
185, 17biadanii 819 . 2 (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ βˆͺ ran Fil) ∧ (𝐹 βŠ† 𝒫 𝑋 ∧ (𝐴 ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹))))
191, 3, 183bitr4ri 304 1 (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ βˆͺ ran Fil ∧ 𝐹 βŠ† 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  {crab 3431   βŠ† wss 3948  π’« cpw 4602  {csn 4628  βˆͺ cuni 4908  ran crn 5677  β€˜cfv 6543  (class class class)co 7412  Topctop 22616  neicnei 22822  Filcfil 23570   fLim cflim 23659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  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-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-top 22617  df-flim 23664
This theorem is referenced by:  flimtop  23690  flimneiss  23691  flimelbas  23693  flimfil  23694  elflim  23696
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