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Theorem isfcf 24058
Description: The property of being a cluster point of a function. (Contributed by Jeff Hankins, 24-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
Assertion
Ref Expression
isfcf ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅))))
Distinct variable groups:   𝐴,𝑜   𝑜,𝑠,𝐽   𝑜,𝐿,𝑠   𝑜,𝐹,𝑠   𝑜,𝑋,𝑠   𝑜,𝑌,𝑠
Allowed substitution hint:   𝐴(𝑠)

Proof of Theorem isfcf
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fcfval 24057 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐽 fClusf 𝐿)‘𝐹) = (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)))
21eleq2d 2825 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ 𝐴 ∈ (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿))))
3 simp1 1135 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝐽 ∈ (TopOn‘𝑋))
4 toponmax 22948 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
5 filfbas 23872 . . . 4 (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
6 id 22 . . . 4 (𝐹:𝑌𝑋𝐹:𝑌𝑋)
7 fmfil 23968 . . . 4 ((𝑋𝐽𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
84, 5, 6, 7syl3an 1159 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
9 fclsopn 24038 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅))))
103, 8, 9syl2anc 584 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅))))
11 simpll1 1211 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) ∧ 𝑠𝐿) → 𝐽 ∈ (TopOn‘𝑋))
1211, 4syl 17 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) ∧ 𝑠𝐿) → 𝑋𝐽)
13 simpll2 1212 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) ∧ 𝑠𝐿) → 𝐿 ∈ (Fil‘𝑌))
1413, 5syl 17 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) ∧ 𝑠𝐿) → 𝐿 ∈ (fBas‘𝑌))
15 simpll3 1213 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) ∧ 𝑠𝐿) → 𝐹:𝑌𝑋)
16 simpl2 1191 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → 𝐿 ∈ (Fil‘𝑌))
17 fgfil 23899 . . . . . . . . . . . 12 (𝐿 ∈ (Fil‘𝑌) → (𝑌filGen𝐿) = 𝐿)
1816, 17syl 17 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (𝑌filGen𝐿) = 𝐿)
1918eleq2d 2825 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (𝑠 ∈ (𝑌filGen𝐿) ↔ 𝑠𝐿))
2019biimpar 477 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) ∧ 𝑠𝐿) → 𝑠 ∈ (𝑌filGen𝐿))
21 eqid 2735 . . . . . . . . . 10 (𝑌filGen𝐿) = (𝑌filGen𝐿)
2221imaelfm 23975 . . . . . . . . 9 (((𝑋𝐽𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑠 ∈ (𝑌filGen𝐿)) → (𝐹𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐿))
2312, 14, 15, 20, 22syl31anc 1372 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) ∧ 𝑠𝐿) → (𝐹𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐿))
24 ineq2 4222 . . . . . . . . . 10 (𝑥 = (𝐹𝑠) → (𝑜𝑥) = (𝑜 ∩ (𝐹𝑠)))
2524neeq1d 2998 . . . . . . . . 9 (𝑥 = (𝐹𝑠) → ((𝑜𝑥) ≠ ∅ ↔ (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
2625rspcv 3618 . . . . . . . 8 ((𝐹𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐿) → (∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅ → (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
2723, 26syl 17 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) ∧ 𝑠𝐿) → (∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅ → (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
2827ralrimdva 3152 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅ → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
29 elfm 23971 . . . . . . . . . . 11 ((𝑋𝐽𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑥𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥)))
304, 5, 6, 29syl3an 1159 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑥𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥)))
3130adantr 480 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑥𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥)))
3231simplbda 499 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) ∧ 𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)) → ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥)
33 r19.29r 3114 . . . . . . . . . 10 ((∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥 ∧ ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → ∃𝑠𝐿 ((𝐹𝑠) ⊆ 𝑥 ∧ (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
34 sslin 4251 . . . . . . . . . . . 12 ((𝐹𝑠) ⊆ 𝑥 → (𝑜 ∩ (𝐹𝑠)) ⊆ (𝑜𝑥))
35 ssn0 4410 . . . . . . . . . . . 12 (((𝑜 ∩ (𝐹𝑠)) ⊆ (𝑜𝑥) ∧ (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → (𝑜𝑥) ≠ ∅)
3634, 35sylan 580 . . . . . . . . . . 11 (((𝐹𝑠) ⊆ 𝑥 ∧ (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → (𝑜𝑥) ≠ ∅)
3736rexlimivw 3149 . . . . . . . . . 10 (∃𝑠𝐿 ((𝐹𝑠) ⊆ 𝑥 ∧ (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → (𝑜𝑥) ≠ ∅)
3833, 37syl 17 . . . . . . . . 9 ((∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥 ∧ ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → (𝑜𝑥) ≠ ∅)
3938ex 412 . . . . . . . 8 (∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥 → (∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅ → (𝑜𝑥) ≠ ∅))
4032, 39syl 17 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) ∧ 𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)) → (∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅ → (𝑜𝑥) ≠ ∅))
4140ralrimdva 3152 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅ → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅))
4228, 41impbid 212 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅ ↔ ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
4342imbi2d 340 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → ((𝐴𝑜 → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅) ↔ (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅)))
4443ralbidva 3174 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (∀𝑜𝐽 (𝐴𝑜 → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅) ↔ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅)))
4544anbi2d 630 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅)) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅))))
462, 10, 453bitrd 305 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴𝑋 ∧ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  cin 3962  wss 3963  c0 4339  cima 5692  wf 6559  cfv 6563  (class class class)co 7431  fBascfbas 21370  filGencfg 21371  TopOnctopon 22932  Filcfil 23869   FilMap cfm 23957   fClus cfcls 23960   fClusf cfcf 23961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-fbas 21379  df-fg 21380  df-top 22916  df-topon 22933  df-cld 23043  df-ntr 23044  df-cls 23045  df-fil 23870  df-fm 23962  df-fcls 23965  df-fcf 23966
This theorem is referenced by:  fcfnei  24059
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