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Theorem isflf 22016
 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 22013 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
21eleq2d 2835 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ 𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿))))
3 simp1 1129 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝐽 ∈ (TopOn‘𝑋))
4 toponmax 20950 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
543ad2ant1 1126 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝑋𝐽)
6 filfbas 21871 . . . . 5 (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
763ad2ant2 1127 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝐿 ∈ (fBas‘𝑌))
8 simp3 1131 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝐹:𝑌𝑋)
9 fmfil 21967 . . . 4 ((𝑋𝐽𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
105, 7, 8, 9syl3anc 1475 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
11 flimopn 21998 . . 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 565 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿)))))
13 elfm 21970 . . . . . . . 8 ((𝑋𝐽𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑜𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜)))
145, 7, 8, 13syl3anc 1475 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑜𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜)))
1514adantr 466 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑜𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜)))
16 toponss 20951 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑜𝐽) → 𝑜𝑋)
173, 16sylan 561 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → 𝑜𝑋)
1817biantrurd 516 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜 ↔ (𝑜𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜)))
1915, 18bitr4d 271 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜))
2019imbi2d 329 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → ((𝐴𝑜𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿)) ↔ (𝐴𝑜 → ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜)))
2120ralbidva 3133 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (∀𝑜𝐽 (𝐴𝑜𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿)) ↔ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜)))
2221anbi2d 606 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜𝑜 ∈ ((𝑋 FilMap 𝐹)‘𝐿))) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑜))))
232, 12, 223bitrd 294 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 196   ∧ wa 382   ∧ w3a 1070   ∈ wcel 2144  ∀wral 3060  ∃wrex 3061   ⊆ wss 3721   “ cima 5252  ⟶wf 6027  ‘cfv 6031  (class class class)co 6792  fBascfbas 19948  TopOnctopon 20934  Filcfil 21868   FilMap cfm 21956   fLim cflim 21957   fLimf cflf 21958 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-map 8010  df-fbas 19957  df-fg 19958  df-top 20918  df-topon 20935  df-ntr 21044  df-nei 21122  df-fil 21869  df-fm 21961  df-flim 21962  df-flf 21963 This theorem is referenced by:  flfelbas  22017  flffbas  22018  flftg  22019  cnpflfi  22022  cnpflf2  22023  txflf  22029  limcflf  23864  rrhre  30399
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