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Theorem isflf 22612
 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 22609 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
21eleq2d 2875 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ 𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿))))
3 simp1 1133 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝐽 ∈ (TopOn‘𝑋))
4 toponmax 21545 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
543ad2ant1 1130 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝑋𝐽)
6 filfbas 22467 . . . . 5 (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
763ad2ant2 1131 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝐿 ∈ (fBas‘𝑌))
8 simp3 1135 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝐹:𝑌𝑋)
9 fmfil 22563 . . . 4 ((𝑋𝐽𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
105, 7, 8, 9syl3anc 1368 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
11 flimopn 22594 . . 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 587 . 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 21546 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜𝐽) → 𝑜𝑋)
143, 13sylan 583 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → 𝑜𝑋)
15 elfm 22566 . . . . . . . 8 ((𝑋𝐽𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑜𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜)))
165, 7, 8, 15syl3anc 1368 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑜𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜)))
1716adantr 484 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑜𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜)))
1814, 17mpbirand 706 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜))
1918imbi2d 344 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → ((𝐴𝑜𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿)) ↔ (𝐴𝑜 → ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜)))
2019ralbidva 3161 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (∀𝑜𝐽 (𝐴𝑜𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿)) ↔ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜)))
2120anbi2d 631 . 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 308 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 209   ∧ wa 399   ∧ w3a 1084   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ⊆ wss 3881   “ cima 5523  ⟶wf 6323  ‘cfv 6327  (class class class)co 7140  fBascfbas 20087  TopOnctopon 21529  Filcfil 22464   FilMap cfm 22552   fLim cflim 22553   fLimf cflf 22554 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-ov 7143  df-oprab 7144  df-mpo 7145  df-map 8398  df-fbas 20096  df-fg 20097  df-top 21513  df-topon 21530  df-ntr 21639  df-nei 21717  df-fil 22465  df-fm 22557  df-flim 22558  df-flf 22559 This theorem is referenced by:  flfelbas  22613  flffbas  22614  flftg  22615  cnpflfi  22618  cnpflf2  22619  txflf  22625  limcflf  24498  rrhre  31408
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