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Theorem flimopn 23699
Description: The condition for being a limit point of a filter still holds if one only considers open neighborhoods. (Contributed by Jeff Hankins, 4-Sep-2009.) (Proof shortened by Mario Carneiro, 9-Apr-2015.)
Assertion
Ref Expression
flimopn ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ βˆ€π‘₯ ∈ 𝐽 (𝐴 ∈ π‘₯ β†’ π‘₯ ∈ 𝐹))))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐹   π‘₯,𝐽   π‘₯,𝑋

Proof of Theorem flimopn
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elflim 23695 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹)))
2 dfss3 3969 . . . 4 (((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹 ↔ βˆ€π‘¦ ∈ ((neiβ€˜π½)β€˜{𝐴})𝑦 ∈ 𝐹)
3 topontop 22635 . . . . . . . . . . 11 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
43ad2antrr 722 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) β†’ 𝐽 ∈ Top)
5 opnneip 22843 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ π‘₯ ∈ 𝐽 ∧ 𝐴 ∈ π‘₯) β†’ π‘₯ ∈ ((neiβ€˜π½)β€˜{𝐴}))
653expb 1118 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (π‘₯ ∈ 𝐽 ∧ 𝐴 ∈ π‘₯)) β†’ π‘₯ ∈ ((neiβ€˜π½)β€˜{𝐴}))
74, 6sylan 578 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ (π‘₯ ∈ 𝐽 ∧ 𝐴 ∈ π‘₯)) β†’ π‘₯ ∈ ((neiβ€˜π½)β€˜{𝐴}))
8 eleq1 2819 . . . . . . . . . 10 (𝑦 = π‘₯ β†’ (𝑦 ∈ 𝐹 ↔ π‘₯ ∈ 𝐹))
98rspcv 3607 . . . . . . . . 9 (π‘₯ ∈ ((neiβ€˜π½)β€˜{𝐴}) β†’ (βˆ€π‘¦ ∈ ((neiβ€˜π½)β€˜{𝐴})𝑦 ∈ 𝐹 β†’ π‘₯ ∈ 𝐹))
107, 9syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ (π‘₯ ∈ 𝐽 ∧ 𝐴 ∈ π‘₯)) β†’ (βˆ€π‘¦ ∈ ((neiβ€˜π½)β€˜{𝐴})𝑦 ∈ 𝐹 β†’ π‘₯ ∈ 𝐹))
1110expr 455 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ π‘₯ ∈ 𝐽) β†’ (𝐴 ∈ π‘₯ β†’ (βˆ€π‘¦ ∈ ((neiβ€˜π½)β€˜{𝐴})𝑦 ∈ 𝐹 β†’ π‘₯ ∈ 𝐹)))
1211com23 86 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ π‘₯ ∈ 𝐽) β†’ (βˆ€π‘¦ ∈ ((neiβ€˜π½)β€˜{𝐴})𝑦 ∈ 𝐹 β†’ (𝐴 ∈ π‘₯ β†’ π‘₯ ∈ 𝐹)))
1312ralrimdva 3152 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) β†’ (βˆ€π‘¦ ∈ ((neiβ€˜π½)β€˜{𝐴})𝑦 ∈ 𝐹 β†’ βˆ€π‘₯ ∈ 𝐽 (𝐴 ∈ π‘₯ β†’ π‘₯ ∈ 𝐹)))
14 simpr 483 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴}))
153ad3antrrr 726 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ 𝐽 ∈ Top)
16 simplr 765 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ 𝐴 ∈ 𝑋)
17 toponuni 22636 . . . . . . . . . . . . . 14 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
1817ad3antrrr 726 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ 𝑋 = βˆͺ 𝐽)
1916, 18eleqtrd 2833 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ 𝐴 ∈ βˆͺ 𝐽)
2019snssd 4811 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ {𝐴} βŠ† βˆͺ 𝐽)
21 eqid 2730 . . . . . . . . . . . . 13 βˆͺ 𝐽 = βˆͺ 𝐽
2221neii1 22830 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ 𝑦 βŠ† βˆͺ 𝐽)
234, 22sylan 578 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ 𝑦 βŠ† βˆͺ 𝐽)
2421neiint 22828 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ {𝐴} βŠ† βˆͺ 𝐽 ∧ 𝑦 βŠ† βˆͺ 𝐽) β†’ (𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴}) ↔ {𝐴} βŠ† ((intβ€˜π½)β€˜π‘¦)))
2515, 20, 23, 24syl3anc 1369 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ (𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴}) ↔ {𝐴} βŠ† ((intβ€˜π½)β€˜π‘¦)))
2614, 25mpbid 231 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ {𝐴} βŠ† ((intβ€˜π½)β€˜π‘¦))
27 snssg 4786 . . . . . . . . . 10 (𝐴 ∈ 𝑋 β†’ (𝐴 ∈ ((intβ€˜π½)β€˜π‘¦) ↔ {𝐴} βŠ† ((intβ€˜π½)β€˜π‘¦)))
2827ad2antlr 723 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ (𝐴 ∈ ((intβ€˜π½)β€˜π‘¦) ↔ {𝐴} βŠ† ((intβ€˜π½)β€˜π‘¦)))
2926, 28mpbird 256 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ 𝐴 ∈ ((intβ€˜π½)β€˜π‘¦))
3021ntropn 22773 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑦 βŠ† βˆͺ 𝐽) β†’ ((intβ€˜π½)β€˜π‘¦) ∈ 𝐽)
3115, 23, 30syl2anc 582 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ ((intβ€˜π½)β€˜π‘¦) ∈ 𝐽)
32 eleq2 2820 . . . . . . . . . . 11 (π‘₯ = ((intβ€˜π½)β€˜π‘¦) β†’ (𝐴 ∈ π‘₯ ↔ 𝐴 ∈ ((intβ€˜π½)β€˜π‘¦)))
33 eleq1 2819 . . . . . . . . . . 11 (π‘₯ = ((intβ€˜π½)β€˜π‘¦) β†’ (π‘₯ ∈ 𝐹 ↔ ((intβ€˜π½)β€˜π‘¦) ∈ 𝐹))
3432, 33imbi12d 343 . . . . . . . . . 10 (π‘₯ = ((intβ€˜π½)β€˜π‘¦) β†’ ((𝐴 ∈ π‘₯ β†’ π‘₯ ∈ 𝐹) ↔ (𝐴 ∈ ((intβ€˜π½)β€˜π‘¦) β†’ ((intβ€˜π½)β€˜π‘¦) ∈ 𝐹)))
3534rspcv 3607 . . . . . . . . 9 (((intβ€˜π½)β€˜π‘¦) ∈ 𝐽 β†’ (βˆ€π‘₯ ∈ 𝐽 (𝐴 ∈ π‘₯ β†’ π‘₯ ∈ 𝐹) β†’ (𝐴 ∈ ((intβ€˜π½)β€˜π‘¦) β†’ ((intβ€˜π½)β€˜π‘¦) ∈ 𝐹)))
3631, 35syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ (βˆ€π‘₯ ∈ 𝐽 (𝐴 ∈ π‘₯ β†’ π‘₯ ∈ 𝐹) β†’ (𝐴 ∈ ((intβ€˜π½)β€˜π‘¦) β†’ ((intβ€˜π½)β€˜π‘¦) ∈ 𝐹)))
3729, 36mpid 44 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ (βˆ€π‘₯ ∈ 𝐽 (𝐴 ∈ π‘₯ β†’ π‘₯ ∈ 𝐹) β†’ ((intβ€˜π½)β€˜π‘¦) ∈ 𝐹))
38 simpllr 772 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ 𝐹 ∈ (Filβ€˜π‘‹))
3921ntrss2 22781 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑦 βŠ† βˆͺ 𝐽) β†’ ((intβ€˜π½)β€˜π‘¦) βŠ† 𝑦)
4015, 23, 39syl2anc 582 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ ((intβ€˜π½)β€˜π‘¦) βŠ† 𝑦)
4123, 18sseqtrrd 4022 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ 𝑦 βŠ† 𝑋)
42 filss 23577 . . . . . . . . . 10 ((𝐹 ∈ (Filβ€˜π‘‹) ∧ (((intβ€˜π½)β€˜π‘¦) ∈ 𝐹 ∧ 𝑦 βŠ† 𝑋 ∧ ((intβ€˜π½)β€˜π‘¦) βŠ† 𝑦)) β†’ 𝑦 ∈ 𝐹)
43423exp2 1352 . . . . . . . . 9 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (((intβ€˜π½)β€˜π‘¦) ∈ 𝐹 β†’ (𝑦 βŠ† 𝑋 β†’ (((intβ€˜π½)β€˜π‘¦) βŠ† 𝑦 β†’ 𝑦 ∈ 𝐹))))
4443com24 95 . . . . . . . 8 (𝐹 ∈ (Filβ€˜π‘‹) β†’ (((intβ€˜π½)β€˜π‘¦) βŠ† 𝑦 β†’ (𝑦 βŠ† 𝑋 β†’ (((intβ€˜π½)β€˜π‘¦) ∈ 𝐹 β†’ 𝑦 ∈ 𝐹))))
4538, 40, 41, 44syl3c 66 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ (((intβ€˜π½)β€˜π‘¦) ∈ 𝐹 β†’ 𝑦 ∈ 𝐹))
4637, 45syld 47 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ (βˆ€π‘₯ ∈ 𝐽 (𝐴 ∈ π‘₯ β†’ π‘₯ ∈ 𝐹) β†’ 𝑦 ∈ 𝐹))
4746ralrimdva 3152 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) β†’ (βˆ€π‘₯ ∈ 𝐽 (𝐴 ∈ π‘₯ β†’ π‘₯ ∈ 𝐹) β†’ βˆ€π‘¦ ∈ ((neiβ€˜π½)β€˜{𝐴})𝑦 ∈ 𝐹))
4813, 47impbid 211 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) β†’ (βˆ€π‘¦ ∈ ((neiβ€˜π½)β€˜{𝐴})𝑦 ∈ 𝐹 ↔ βˆ€π‘₯ ∈ 𝐽 (𝐴 ∈ π‘₯ β†’ π‘₯ ∈ 𝐹)))
492, 48bitrid 282 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) ∧ 𝐴 ∈ 𝑋) β†’ (((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹 ↔ βˆ€π‘₯ ∈ 𝐽 (𝐴 ∈ π‘₯ β†’ π‘₯ ∈ 𝐹)))
5049pm5.32da 577 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ ((𝐴 ∈ 𝑋 ∧ ((neiβ€˜π½)β€˜{𝐴}) βŠ† 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ βˆ€π‘₯ ∈ 𝐽 (𝐴 ∈ π‘₯ β†’ π‘₯ ∈ 𝐹))))
511, 50bitrd 278 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 ∈ (Filβ€˜π‘‹)) β†’ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ βˆ€π‘₯ ∈ 𝐽 (𝐴 ∈ π‘₯ β†’ π‘₯ ∈ 𝐹))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059   βŠ† wss 3947  {csn 4627  βˆͺ cuni 4907  β€˜cfv 6542  (class class class)co 7411  Topctop 22615  TopOnctopon 22632  intcnt 22741  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-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
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-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  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-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-fbas 21141  df-top 22616  df-topon 22633  df-ntr 22744  df-nei 22822  df-fil 23570  df-flim 23663
This theorem is referenced by:  fbflim  23700  flimrest  23707  flimsncls  23710  isflf  23717  cnpflfi  23723  flimfnfcls  23752  alexsublem  23768  cfilfcls  25022  iscmet3lem2  25040
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