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Theorem isflf 24017
Description: The property of being a limit point of a function. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
Assertion
Ref Expression
isflf ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜))))
Distinct variable groups:   𝐴,𝑜   𝑜,𝑠,𝐹   𝑜,𝐽,𝑠   𝑜,𝐿,𝑠   𝑜,𝑋,𝑠   𝑜,𝑌,𝑠
Allowed substitution hint:   𝐴(𝑠)

Proof of Theorem isflf
StepHypRef Expression
1 flfval 24014 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
21eleq2d 2825 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ 𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿))))
3 simp1 1135 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝐽 ∈ (TopOn‘𝑋))
4 toponmax 22948 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
543ad2ant1 1132 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝑋𝐽)
6 filfbas 23872 . . . . 5 (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
763ad2ant2 1133 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝐿 ∈ (fBas‘𝑌))
8 simp3 1137 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝐹:𝑌𝑋)
9 fmfil 23968 . . . 4 ((𝑋𝐽𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
105, 7, 8, 9syl3anc 1370 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
11 flimopn 23999 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿)))))
123, 10, 11syl2anc 584 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿)))))
13 toponss 22949 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜𝐽) → 𝑜𝑋)
143, 13sylan 580 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → 𝑜𝑋)
15 elfm 23971 . . . . . . . 8 ((𝑋𝐽𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑜𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜)))
165, 7, 8, 15syl3anc 1370 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑜𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜)))
1716adantr 480 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑜𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜)))
1814, 17mpbirand 707 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜))
1918imbi2d 340 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → ((𝐴𝑜𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿)) ↔ (𝐴𝑜 → ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜)))
2019ralbidva 3174 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (∀𝑜𝐽 (𝐴𝑜𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿)) ↔ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜)))
2120anbi2d 630 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿))) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜))))
222, 12, 213bitrd 305 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086  wcel 2106  wral 3059  wrex 3068  wss 3963  cima 5692  wf 6559  cfv 6563  (class class class)co 7431  fBascfbas 21370  TopOnctopon 22932  Filcfil 23869   FilMap cfm 23957   fLim cflim 23958   fLimf cflf 23959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-fbas 21379  df-fg 21380  df-top 22916  df-topon 22933  df-ntr 23044  df-nei 23122  df-fil 23870  df-fm 23962  df-flim 23963  df-flf 23964
This theorem is referenced by:  flfelbas  24018  flffbas  24019  flftg  24020  cnpflfi  24023  cnpflf2  24024  txflf  24030  limcflf  25931  rrhre  33984
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