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Theorem flffbas 23943
Description: Limit points of a function can be defined using filter bases. (Contributed by Jeff Hankins, 9-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.)
Hypothesis
Ref Expression
flffbas.l 𝐿 = (𝑌filGen𝐵)
Assertion
Ref Expression
flffbas ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜))))
Distinct variable groups:   𝑜,𝑠,𝐴   𝐵,𝑜,𝑠   𝑜,𝐹,𝑠   𝑜,𝐽,𝑠   𝑜,𝐿,𝑠   𝑜,𝑋,𝑠   𝑜,𝑌,𝑠

Proof of Theorem flffbas
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 flffbas.l . . . 4 𝐿 = (𝑌filGen𝐵)
2 fgcl 23826 . . . 4 (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌))
31, 2eqeltrid 2829 . . 3 (𝐵 ∈ (fBas‘𝑌) → 𝐿 ∈ (Fil‘𝑌))
4 isflf 23941 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜))))
53, 4syl3an2 1161 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜))))
61eleq2i 2817 . . . . . . . 8 (𝑡𝐿𝑡 ∈ (𝑌filGen𝐵))
7 elfg 23819 . . . . . . . . . . 11 (𝐵 ∈ (fBas‘𝑌) → (𝑡 ∈ (𝑌filGen𝐵) ↔ (𝑡𝑌 ∧ ∃𝑠𝐵 𝑠𝑡)))
873ad2ant2 1131 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑡 ∈ (𝑌filGen𝐵) ↔ (𝑡𝑌 ∧ ∃𝑠𝐵 𝑠𝑡)))
9 sstr2 3983 . . . . . . . . . . . . . . . 16 ((𝐹𝑠) ⊆ (𝐹𝑡) → ((𝐹𝑡) ⊆ 𝑜 → (𝐹𝑠) ⊆ 𝑜))
10 imass2 6107 . . . . . . . . . . . . . . . 16 (𝑠𝑡 → (𝐹𝑠) ⊆ (𝐹𝑡))
119, 10syl11 33 . . . . . . . . . . . . . . 15 ((𝐹𝑡) ⊆ 𝑜 → (𝑠𝑡 → (𝐹𝑠) ⊆ 𝑜))
1211adantl 480 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐹𝑡) ⊆ 𝑜) → (𝑠𝑡 → (𝐹𝑠) ⊆ 𝑜))
1312reximdv 3159 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐹𝑡) ⊆ 𝑜) → (∃𝑠𝐵 𝑠𝑡 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜))
1413ex 411 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐹𝑡) ⊆ 𝑜 → (∃𝑠𝐵 𝑠𝑡 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜)))
1514com23 86 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (∃𝑠𝐵 𝑠𝑡 → ((𝐹𝑡) ⊆ 𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜)))
1615adantld 489 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑡𝑌 ∧ ∃𝑠𝐵 𝑠𝑡) → ((𝐹𝑡) ⊆ 𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜)))
178, 16sylbid 239 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑡 ∈ (𝑌filGen𝐵) → ((𝐹𝑡) ⊆ 𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜)))
1817adantr 479 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (𝑡 ∈ (𝑌filGen𝐵) → ((𝐹𝑡) ⊆ 𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜)))
196, 18biimtrid 241 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (𝑡𝐿 → ((𝐹𝑡) ⊆ 𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜)))
2019rexlimdv 3142 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜))
21 ssfg 23820 . . . . . . . . . . . 12 (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ (𝑌filGen𝐵))
2221, 1sseqtrrdi 4028 . . . . . . . . . . 11 (𝐵 ∈ (fBas‘𝑌) → 𝐵𝐿)
2322sselda 3976 . . . . . . . . . 10 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝑠𝐵) → 𝑠𝐿)
24233ad2antl2 1183 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑠𝐵) → 𝑠𝐿)
2524ad2ant2r 745 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ (𝑠𝐵 ∧ (𝐹𝑠) ⊆ 𝑜)) → 𝑠𝐿)
26 simprr 771 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ (𝑠𝐵 ∧ (𝐹𝑠) ⊆ 𝑜)) → (𝐹𝑠) ⊆ 𝑜)
27 imaeq2 6060 . . . . . . . . . 10 (𝑡 = 𝑠 → (𝐹𝑡) = (𝐹𝑠))
2827sseq1d 4008 . . . . . . . . 9 (𝑡 = 𝑠 → ((𝐹𝑡) ⊆ 𝑜 ↔ (𝐹𝑠) ⊆ 𝑜))
2928rspcev 3606 . . . . . . . 8 ((𝑠𝐿 ∧ (𝐹𝑠) ⊆ 𝑜) → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜)
3025, 26, 29syl2anc 582 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ (𝑠𝐵 ∧ (𝐹𝑠) ⊆ 𝑜)) → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜)
3130rexlimdvaa 3145 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜 → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜))
3220, 31impbid 211 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜 ↔ ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜))
3332imbi2d 339 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → ((𝐴𝑜 → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜) ↔ (𝐴𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜)))
3433ralbidv 3167 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (∀𝑜𝐽 (𝐴𝑜 → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜) ↔ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜)))
3534pm5.32da 577 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜)) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜))))
365, 35bitrd 278 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3050  wrex 3059  wss 3944  cima 5681  wf 6545  cfv 6549  (class class class)co 7419  fBascfbas 21284  filGencfg 21285  TopOnctopon 22856  Filcfil 23793   fLimf cflf 23883
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 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-map 8847  df-fbas 21293  df-fg 21294  df-top 22840  df-topon 22857  df-ntr 22968  df-nei 23046  df-fil 23794  df-fm 23886  df-flim 23887  df-flf 23888
This theorem is referenced by:  lmflf  23953  eltsms  24081
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