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Theorem hausflimlem 23903
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 2728 . . . 4 βˆͺ 𝐽 = βˆͺ 𝐽
32flimfil 23893 . . 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 23889 . . . . 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 23043 . . . 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 23891 . . 3 ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ π‘ˆ ∈ ((neiβ€˜π½)β€˜{𝐴})) β†’ π‘ˆ ∈ 𝐹)
121, 10, 11syl2anc 582 . 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 23043 . . . 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 23891 . . 3 ((𝐡 ∈ (𝐽 fLim 𝐹) ∧ 𝑉 ∈ ((neiβ€˜π½)β€˜{𝐡})) β†’ 𝑉 ∈ 𝐹)
1913, 17, 18syl2anc 582 . 2 (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐡 ∈ (𝐽 fLim 𝐹)) ∧ (π‘ˆ ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ 𝑉)) β†’ 𝑉 ∈ 𝐹)
20 filinn0 23784 . 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 394   ∧ w3a 1084   ∈ wcel 2098   β‰  wne 2937   ∩ cin 3948  βˆ…c0 4326  {csn 4632  βˆͺ cuni 4912  β€˜cfv 6553  (class class class)co 7426  Topctop 22815  neicnei 23021  Filcfil 23769   fLim cflim 23858
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-fbas 21283  df-top 22816  df-nei 23022  df-fil 23770  df-flim 23863
This theorem is referenced by:  hausflimi  23904
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