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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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