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Theorem elflim2 23688
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 467 . 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 1087 . . 3 ((𝐽 ∈ Top ∧ 𝐹 ∈ βˆͺ ran Fil ∧ 𝐹 βŠ† 𝒫 𝑋) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ βˆͺ ran Fil) ∧ 𝐹 βŠ† 𝒫 𝑋))
32anbi1i 622 . 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 23663 . . . 4 fLim = (𝑗 ∈ Top, 𝑓 ∈ βˆͺ ran Fil ↦ {π‘₯ ∈ βˆͺ 𝑗 ∣ (((neiβ€˜π‘—)β€˜{π‘₯}) βŠ† 𝑓 ∧ 𝑓 βŠ† 𝒫 βˆͺ 𝑗)})
54elmpocl 7650 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ (𝐽 ∈ Top ∧ 𝐹 ∈ βˆͺ ran Fil))
6 flimval.1 . . . . . 6 𝑋 = βˆͺ 𝐽
76flimval 23687 . . . . 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 2817 . . . 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 4637 . . . . . . . . . 10 (π‘₯ = 𝐴 β†’ {π‘₯} = {𝐴})
109fveq2d 6894 . . . . . . . . 9 (π‘₯ = 𝐴 β†’ ((neiβ€˜π½)β€˜{π‘₯}) = ((neiβ€˜π½)β€˜{𝐴}))
1110sseq1d 4012 . . . . . . . 8 (π‘₯ = 𝐴 β†’ (((neiβ€˜π½)β€˜{π‘₯}) βŠ† 𝐹 ↔ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹))
1211anbi1d 628 . . . . . . 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 462 . . . . . 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 3682 . . . . 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 641 . . . . 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 274 . . . 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 286 . . 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 818 . 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 303 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  {crab 3430   βŠ† wss 3947  π’« cpw 4601  {csn 4627  βˆͺ cuni 4907  ran crn 5676  β€˜cfv 6542  (class class class)co 7411  Topctop 22615  neicnei 22821  Filcfil 23569   fLim cflim 23658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-top 22616  df-flim 23663
This theorem is referenced by:  flimtop  23689  flimneiss  23690  flimelbas  23692  flimfil  23693  elflim  23695
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