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Theorem flffbas 22020
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 21903 . . . 4 (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌))
31, 2syl5eqel 2843 . . 3 (𝐵 ∈ (fBas‘𝑌) → 𝐿 ∈ (Fil‘𝑌))
4 isflf 22018 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜))))
53, 4syl3an2 1168 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜))))
61eleq2i 2831 . . . . . . . 8 (𝑡𝐿𝑡 ∈ (𝑌filGen𝐵))
7 elfg 21896 . . . . . . . . . . 11 (𝐵 ∈ (fBas‘𝑌) → (𝑡 ∈ (𝑌filGen𝐵) ↔ (𝑡𝑌 ∧ ∃𝑠𝐵 𝑠𝑡)))
873ad2ant2 1129 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑡 ∈ (𝑌filGen𝐵) ↔ (𝑡𝑌 ∧ ∃𝑠𝐵 𝑠𝑡)))
9 sstr2 3751 . . . . . . . . . . . . . . . 16 ((𝐹𝑠) ⊆ (𝐹𝑡) → ((𝐹𝑡) ⊆ 𝑜 → (𝐹𝑠) ⊆ 𝑜))
10 imass2 5659 . . . . . . . . . . . . . . . 16 (𝑠𝑡 → (𝐹𝑠) ⊆ (𝐹𝑡))
119, 10syl11 33 . . . . . . . . . . . . . . 15 ((𝐹𝑡) ⊆ 𝑜 → (𝑠𝑡 → (𝐹𝑠) ⊆ 𝑜))
1211adantl 473 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐹𝑡) ⊆ 𝑜) → (𝑠𝑡 → (𝐹𝑠) ⊆ 𝑜))
1312reximdv 3154 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐹𝑡) ⊆ 𝑜) → (∃𝑠𝐵 𝑠𝑡 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜))
1413ex 449 . . . . . . . . . . . 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 484 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑡𝑌 ∧ ∃𝑠𝐵 𝑠𝑡) → ((𝐹𝑡) ⊆ 𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜)))
178, 16sylbid 230 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑡 ∈ (𝑌filGen𝐵) → ((𝐹𝑡) ⊆ 𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜)))
1817adantr 472 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (𝑡 ∈ (𝑌filGen𝐵) → ((𝐹𝑡) ⊆ 𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜)))
196, 18syl5bi 232 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (𝑡𝐿 → ((𝐹𝑡) ⊆ 𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜)))
2019rexlimdv 3168 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜))
21 ssfg 21897 . . . . . . . . . . . 12 (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ (𝑌filGen𝐵))
2221, 1syl6sseqr 3793 . . . . . . . . . . 11 (𝐵 ∈ (fBas‘𝑌) → 𝐵𝐿)
2322sselda 3744 . . . . . . . . . 10 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝑠𝐵) → 𝑠𝐿)
24233ad2antl2 1202 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑠𝐵) → 𝑠𝐿)
2524ad2ant2r 800 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ (𝑠𝐵 ∧ (𝐹𝑠) ⊆ 𝑜)) → 𝑠𝐿)
26 simprr 813 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ (𝑠𝐵 ∧ (𝐹𝑠) ⊆ 𝑜)) → (𝐹𝑠) ⊆ 𝑜)
27 imaeq2 5620 . . . . . . . . . 10 (𝑡 = 𝑠 → (𝐹𝑡) = (𝐹𝑠))
2827sseq1d 3773 . . . . . . . . 9 (𝑡 = 𝑠 → ((𝐹𝑡) ⊆ 𝑜 ↔ (𝐹𝑠) ⊆ 𝑜))
2928rspcev 3449 . . . . . . . 8 ((𝑠𝐿 ∧ (𝐹𝑠) ⊆ 𝑜) → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜)
3025, 26, 29syl2anc 696 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ (𝑠𝐵 ∧ (𝐹𝑠) ⊆ 𝑜)) → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜)
3130rexlimdvaa 3170 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜 → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜))
3220, 31impbid 202 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜 ↔ ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜))
3332imbi2d 329 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → ((𝐴𝑜 → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜) ↔ (𝐴𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜)))
3433ralbidv 3124 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (∀𝑜𝐽 (𝐴𝑜 → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜) ↔ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜)))
3534pm5.32da 676 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜)) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜))))
365, 35bitrd 268 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wrex 3051  wss 3715  cima 5269  wf 6045  cfv 6049  (class class class)co 6814  fBascfbas 19956  filGencfg 19957  TopOnctopon 20937  Filcfil 21870   fLimf cflf 21960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-map 8027  df-fbas 19965  df-fg 19966  df-top 20921  df-topon 20938  df-ntr 21046  df-nei 21124  df-fil 21871  df-fm 21963  df-flim 21964  df-flf 21965
This theorem is referenced by:  lmflf  22030  eltsms  22157
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