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Theorem flfnei 21700
Description: The property of being a limit point of a function in terms of neighborhoods. (Contributed by Jeff Hankins, 9-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Assertion
Ref Expression
flfnei ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑛)))
Distinct variable groups:   𝑛,𝑠,𝐹   𝐴,𝑛   𝑛,𝐽,𝑠   𝑛,𝐿,𝑠   𝑛,𝑋,𝑠   𝑛,𝑌,𝑠
Allowed substitution hint:   𝐴(𝑠)

Proof of Theorem flfnei
StepHypRef Expression
1 flfval 21699 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
21eleq2d 2689 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ 𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿))))
3 simp1 1059 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝐽 ∈ (TopOn‘𝑋))
4 toponmax 20638 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
543ad2ant1 1080 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝑋𝐽)
6 filfbas 21557 . . . . 5 (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
763ad2ant2 1081 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝐿 ∈ (fBas‘𝑌))
8 simp3 1061 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝐹:𝑌𝑋)
9 fmfil 21653 . . . 4 ((𝑋𝐽𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
105, 7, 8, 9syl3anc 1323 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
11 elflim 21680 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)) ↔ (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ ((𝑋 FilMap 𝐹)‘𝐿))))
123, 10, 11syl2anc 692 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)) ↔ (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ ((𝑋 FilMap 𝐹)‘𝐿))))
13 dfss3 3578 . . . 4 (((nei‘𝐽)‘{𝐴}) ⊆ ((𝑋 FilMap 𝐹)‘𝐿) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})𝑛 ∈ ((𝑋 FilMap 𝐹)‘𝐿))
14 topontop 20636 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
15143ad2ant1 1080 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝐽 ∈ Top)
16 eqid 2626 . . . . . . . . 9 𝐽 = 𝐽
1716neii1 20815 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑛 𝐽)
1815, 17sylan 488 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑛 𝐽)
19 toponuni 20637 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
20193ad2ant1 1080 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝑋 = 𝐽)
2120adantr 481 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑋 = 𝐽)
2218, 21sseqtr4d 3626 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑛𝑋)
23 elfm 21656 . . . . . . . 8 ((𝑋𝐽𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑛 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑛𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑛)))
245, 7, 8, 23syl3anc 1323 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑛 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑛𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑛)))
2524baibd 947 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑛𝑋) → (𝑛 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑛))
2622, 25syldan 487 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → (𝑛 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑛))
2726ralbidva 2984 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})𝑛 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑛))
2813, 27syl5bb 272 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (((nei‘𝐽)‘{𝐴}) ⊆ ((𝑋 FilMap 𝐹)‘𝐿) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑛))
2928anbi2d 739 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ ((𝑋 FilMap 𝐹)‘𝐿)) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑛)))
302, 12, 293bitrd 294 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑛)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  wral 2912  wrex 2913  wss 3560  {csn 4153   cuni 4407  cima 5082  wf 5846  cfv 5850  (class class class)co 6605  fBascfbas 19648  Topctop 20612  TopOnctopon 20613  neicnei 20806  Filcfil 21554   FilMap cfm 21642   fLim cflim 21643   fLimf cflf 21644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-map 7805  df-fbas 19657  df-fg 19658  df-top 20616  df-topon 20618  df-nei 20807  df-fil 21555  df-fm 21647  df-flim 21648  df-flf 21649
This theorem is referenced by:  flfneii  21701  cnextcn  21776  cnextfres1  21777
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