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Theorem flffbas 22606
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 22489 . . . 4 (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌))
31, 2eqeltrid 2920 . . 3 (𝐵 ∈ (fBas‘𝑌) → 𝐿 ∈ (Fil‘𝑌))
4 isflf 22604 . . 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 2907 . . . . . . . 8 (𝑡𝐿𝑡 ∈ (𝑌filGen𝐵))
7 elfg 22482 . . . . . . . . . . 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 3960 . . . . . . . . . . . . . . . 16 ((𝐹𝑠) ⊆ (𝐹𝑡) → ((𝐹𝑡) ⊆ 𝑜 → (𝐹𝑠) ⊆ 𝑜))
10 imass2 5952 . . . . . . . . . . . . . . . 16 (𝑠𝑡 → (𝐹𝑠) ⊆ (𝐹𝑡))
119, 10syl11 33 . . . . . . . . . . . . . . 15 ((𝐹𝑡) ⊆ 𝑜 → (𝑠𝑡 → (𝐹𝑠) ⊆ 𝑜))
1211adantl 485 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐹𝑡) ⊆ 𝑜) → (𝑠𝑡 → (𝐹𝑠) ⊆ 𝑜))
1312reximdv 3265 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐹𝑡) ⊆ 𝑜) → (∃𝑠𝐵 𝑠𝑡 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜))
1413ex 416 . . . . . . . . . . . 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 494 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑡𝑌 ∧ ∃𝑠𝐵 𝑠𝑡) → ((𝐹𝑡) ⊆ 𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜)))
178, 16sylbid 243 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑡 ∈ (𝑌filGen𝐵) → ((𝐹𝑡) ⊆ 𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜)))
1817adantr 484 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (𝑡 ∈ (𝑌filGen𝐵) → ((𝐹𝑡) ⊆ 𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜)))
196, 18syl5bi 245 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (𝑡𝐿 → ((𝐹𝑡) ⊆ 𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜)))
2019rexlimdv 3275 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜))
21 ssfg 22483 . . . . . . . . . . . 12 (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ (𝑌filGen𝐵))
2221, 1sseqtrrdi 4004 . . . . . . . . . . 11 (𝐵 ∈ (fBas‘𝑌) → 𝐵𝐿)
2322sselda 3953 . . . . . . . . . 10 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝑠𝐵) → 𝑠𝐿)
24233ad2antl2 1183 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑠𝐵) → 𝑠𝐿)
2524ad2ant2r 746 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ (𝑠𝐵 ∧ (𝐹𝑠) ⊆ 𝑜)) → 𝑠𝐿)
26 simprr 772 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ (𝑠𝐵 ∧ (𝐹𝑠) ⊆ 𝑜)) → (𝐹𝑠) ⊆ 𝑜)
27 imaeq2 5912 . . . . . . . . . 10 (𝑡 = 𝑠 → (𝐹𝑡) = (𝐹𝑠))
2827sseq1d 3984 . . . . . . . . 9 (𝑡 = 𝑠 → ((𝐹𝑡) ⊆ 𝑜 ↔ (𝐹𝑠) ⊆ 𝑜))
2928rspcev 3609 . . . . . . . 8 ((𝑠𝐿 ∧ (𝐹𝑠) ⊆ 𝑜) → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜)
3025, 26, 29syl2anc 587 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ (𝑠𝐵 ∧ (𝐹𝑠) ⊆ 𝑜)) → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜)
3130rexlimdvaa 3277 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜 → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜))
3220, 31impbid 215 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜 ↔ ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜))
3332imbi2d 344 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → ((𝐴𝑜 → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜) ↔ (𝐴𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜)))
3433ralbidv 3192 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (∀𝑜𝐽 (𝐴𝑜 → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜) ↔ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜)))
3534pm5.32da 582 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜)) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜))))
365, 35bitrd 282 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  wrex 3134  wss 3919  cima 5545  wf 6339  cfv 6343  (class class class)co 7149  fBascfbas 20086  filGencfg 20087  TopOnctopon 21521  Filcfil 22456   fLimf cflf 22546
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-map 8404  df-fbas 20095  df-fg 20096  df-top 21505  df-topon 21522  df-ntr 21631  df-nei 21709  df-fil 22457  df-fm 22549  df-flim 22550  df-flf 22551
This theorem is referenced by:  lmflf  22616  eltsms  22744
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