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Theorem hausflimlem 23805
Description: If 𝐴 and 𝐡 are both limits of the same filter, then all neighborhoods of 𝐴 and 𝐡 intersect. (Contributed by Mario Carneiro, 21-Sep-2015.)
Assertion
Ref Expression
hausflimlem (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ (π‘ˆ ∩ 𝑉) β‰  βˆ…)

Proof of Theorem hausflimlem
StepHypRef Expression
1 simp1l 1194 . . 3 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ 𝐴 ∈ (𝐽 fLim 𝐹))
2 eqid 2724 . . . 4 βˆͺ 𝐽 = βˆͺ 𝐽
32flimfil 23795 . . 3 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝐹 ∈ (Filβ€˜βˆͺ 𝐽))
41, 3syl 17 . 2 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ 𝐹 ∈ (Filβ€˜βˆͺ 𝐽))
5 flimtop 23791 . . . . 5 (𝐴 ∈ (𝐽 fLim 𝐹) β†’ 𝐽 ∈ Top)
61, 5syl 17 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ 𝐽 ∈ Top)
7 simp2l 1196 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ π‘ˆ ∈ 𝐽)
8 simp3l 1198 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ 𝐴 ∈ π‘ˆ)
9 opnneip 22945 . . . 4 ((𝐽 ∈ Top ∧ π‘ˆ ∈ 𝐽 ∧ 𝐴 ∈ π‘ˆ) β†’ π‘ˆ ∈ ((neiβ€˜π½)β€˜{𝐴}))
106, 7, 8, 9syl3anc 1368 . . 3 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ π‘ˆ ∈ ((neiβ€˜π½)β€˜{𝐴}))
11 flimnei 23793 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘ˆ ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ π‘ˆ ∈ 𝐹)
121, 10, 11syl2anc 583 . 2 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ π‘ˆ ∈ 𝐹)
13 simp1r 1195 . . 3 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ 𝐡 ∈ (𝐽 fLim 𝐹))
14 simp2r 1197 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ 𝑉 ∈ 𝐽)
15 simp3r 1199 . . . 4 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ 𝐡 ∈ 𝑉)
16 opnneip 22945 . . . 4 ((𝐽 ∈ Top ∧ 𝑉 ∈ 𝐽 ∧ 𝐡 ∈ 𝑉) β†’ 𝑉 ∈ ((neiβ€˜π½)β€˜{𝐡}))
176, 14, 15, 16syl3anc 1368 . . 3 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ 𝑉 ∈ ((neiβ€˜π½)β€˜{𝐡}))
18 flimnei 23793 . . 3 ((𝐡 ∈ (𝐽 fLim 𝐹) ∧ 𝑉 ∈ ((neiβ€˜π½)β€˜{𝐡})) β†’ 𝑉 ∈ 𝐹)
1913, 17, 18syl2anc 583 . 2 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ 𝑉 ∈ 𝐹)
20 filinn0 23686 . 2 ((𝐹 ∈ (Filβ€˜βˆͺ 𝐽) ∧ π‘ˆ ∈ 𝐹 ∧ 𝑉 ∈ 𝐹) β†’ (π‘ˆ ∩ 𝑉) β‰  βˆ…)
214, 12, 19, 20syl3anc 1368 1 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ (π‘ˆ ∩ 𝑉) β‰  βˆ…)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   ∈ wcel 2098   β‰  wne 2932   ∩ cin 3939  βˆ…c0 4314  {csn 4620  βˆͺ cuni 4899  β€˜cfv 6533  (class class class)co 7401  Topctop 22717  neicnei 22923  Filcfil 23671   fLim cflim 23760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-fbas 21225  df-top 22718  df-nei 22924  df-fil 23672  df-flim 23765
This theorem is referenced by:  hausflimi  23806
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