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Theorem flffbas 21722
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 21605 . . . 4 (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌))
31, 2syl5eqel 2702 . . 3 (𝐵 ∈ (fBas‘𝑌) → 𝐿 ∈ (Fil‘𝑌))
4 isflf 21720 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜))))
53, 4syl3an2 1357 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜))))
61eleq2i 2690 . . . . . . . 8 (𝑡𝐿𝑡 ∈ (𝑌filGen𝐵))
7 elfg 21598 . . . . . . . . . . 11 (𝐵 ∈ (fBas‘𝑌) → (𝑡 ∈ (𝑌filGen𝐵) ↔ (𝑡𝑌 ∧ ∃𝑠𝐵 𝑠𝑡)))
873ad2ant2 1081 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑡 ∈ (𝑌filGen𝐵) ↔ (𝑡𝑌 ∧ ∃𝑠𝐵 𝑠𝑡)))
9 sstr2 3594 . . . . . . . . . . . . . . . 16 ((𝐹𝑠) ⊆ (𝐹𝑡) → ((𝐹𝑡) ⊆ 𝑜 → (𝐹𝑠) ⊆ 𝑜))
10 imass2 5465 . . . . . . . . . . . . . . . 16 (𝑠𝑡 → (𝐹𝑠) ⊆ (𝐹𝑡))
119, 10syl11 33 . . . . . . . . . . . . . . 15 ((𝐹𝑡) ⊆ 𝑜 → (𝑠𝑡 → (𝐹𝑠) ⊆ 𝑜))
1211adantl 482 . . . . . . . . . . . . . 14 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐹𝑡) ⊆ 𝑜) → (𝑠𝑡 → (𝐹𝑠) ⊆ 𝑜))
1312reximdv 3011 . . . . . . . . . . . . 13 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ (𝐹𝑡) ⊆ 𝑜) → (∃𝑠𝐵 𝑠𝑡 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜))
1413ex 450 . . . . . . . . . . . 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 483 . . . . . . . . . 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 481 . . . . . . . 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 3024 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜))
21 ssfg 21599 . . . . . . . . . . . 12 (𝐵 ∈ (fBas‘𝑌) → 𝐵 ⊆ (𝑌filGen𝐵))
2221, 1syl6sseqr 3636 . . . . . . . . . . 11 (𝐵 ∈ (fBas‘𝑌) → 𝐵𝐿)
2322sselda 3587 . . . . . . . . . 10 ((𝐵 ∈ (fBas‘𝑌) ∧ 𝑠𝐵) → 𝑠𝐿)
24233ad2antl2 1222 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑠𝐵) → 𝑠𝐿)
2524ad2ant2r 782 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ (𝑠𝐵 ∧ (𝐹𝑠) ⊆ 𝑜)) → 𝑠𝐿)
26 simprr 795 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ (𝑠𝐵 ∧ (𝐹𝑠) ⊆ 𝑜)) → (𝐹𝑠) ⊆ 𝑜)
27 imaeq2 5426 . . . . . . . . . 10 (𝑡 = 𝑠 → (𝐹𝑡) = (𝐹𝑠))
2827sseq1d 3616 . . . . . . . . 9 (𝑡 = 𝑠 → ((𝐹𝑡) ⊆ 𝑜 ↔ (𝐹𝑠) ⊆ 𝑜))
2928rspcev 3298 . . . . . . . 8 ((𝑠𝐿 ∧ (𝐹𝑠) ⊆ 𝑜) → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜)
3025, 26, 29syl2anc 692 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) ∧ (𝑠𝐵 ∧ (𝐹𝑠) ⊆ 𝑜)) → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜)
3130rexlimdvaa 3026 . . . . . 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 330 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → ((𝐴𝑜 → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜) ↔ (𝐴𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜)))
3433ralbidv 2981 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝐴𝑋) → (∀𝑜𝐽 (𝐴𝑜 → ∃𝑡𝐿 (𝐹𝑡) ⊆ 𝑜) ↔ ∀𝑜𝐽 (𝐴𝑜 → ∃𝑠𝐵 (𝐹𝑠) ⊆ 𝑜)))
3534pm5.32da 672 . 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  wss 3559  cima 5082  wf 5848  cfv 5852  (class class class)co 6610  fBascfbas 19666  filGencfg 19667  TopOnctopon 20647  Filcfil 21572   fLimf cflf 21662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-map 7811  df-fbas 19675  df-fg 19676  df-top 20631  df-topon 20648  df-ntr 20747  df-nei 20825  df-fil 21573  df-fm 21665  df-flim 21666  df-flf 21667
This theorem is referenced by:  lmflf  21732  eltsms  21859
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