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Theorem isfcf 21885
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 21884 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐽 fClusf 𝐿)‘𝐹) = (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)))
21eleq2d 2716 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ 𝐴 ∈ (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿))))
3 simp1 1081 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝐽 ∈ (TopOn‘𝑋))
4 toponmax 20778 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
5 filfbas 21699 . . . 4 (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
6 id 22 . . . 4 (𝐹:𝑌𝑋𝐹:𝑌𝑋)
7 fmfil 21795 . . . 4 ((𝑋𝐽𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
84, 5, 6, 7syl3an 1408 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
9 fclsopn 21865 . . 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 694 . 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 1120 . . . . . . . . . 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 1121 . . . . . . . . . 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 1122 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) ∧ 𝑠𝐿) → 𝐹:𝑌𝑋)
16 simpl2 1085 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → 𝐿 ∈ (Fil‘𝑌))
17 fgfil 21726 . . . . . . . . . . . 12 (𝐿 ∈ (Fil‘𝑌) → (𝑌filGen𝐿) = 𝐿)
1816, 17syl 17 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (𝑌filGen𝐿) = 𝐿)
1918eleq2d 2716 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (𝑠 ∈ (𝑌filGen𝐿) ↔ 𝑠𝐿))
2019biimpar 501 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) ∧ 𝑠𝐿) → 𝑠 ∈ (𝑌filGen𝐿))
21 eqid 2651 . . . . . . . . . 10 (𝑌filGen𝐿) = (𝑌filGen𝐿)
2221imaelfm 21802 . . . . . . . . 9 (((𝑋𝐽𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑠 ∈ (𝑌filGen𝐿)) → (𝐹𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐿))
2312, 14, 15, 20, 22syl31anc 1369 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) ∧ 𝑠𝐿) → (𝐹𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐿))
24 ineq2 3841 . . . . . . . . . 10 (𝑥 = (𝐹𝑠) → (𝑜𝑥) = (𝑜 ∩ (𝐹𝑠)))
2524neeq1d 2882 . . . . . . . . 9 (𝑥 = (𝐹𝑠) → ((𝑜𝑥) ≠ ∅ ↔ (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
2625rspcv 3336 . . . . . . . 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 2998 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅ → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
29 elfm 21798 . . . . . . . . . . 11 ((𝑋𝐽𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑥𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥)))
304, 5, 6, 29syl3an 1408 . . . . . . . . . 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 653 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) ∧ 𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)) → ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥)
33 r19.29r 3102 . . . . . . . . . 10 ((∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥 ∧ ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → ∃𝑠𝐿 ((𝐹𝑠) ⊆ 𝑥 ∧ (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
34 sslin 3872 . . . . . . . . . . . 12 ((𝐹𝑠) ⊆ 𝑥 → (𝑜 ∩ (𝐹𝑠)) ⊆ (𝑜𝑥))
35 ssn0 4009 . . . . . . . . . . . 12 (((𝑜 ∩ (𝐹𝑠)) ⊆ (𝑜𝑥) ∧ (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → (𝑜𝑥) ≠ ∅)
3634, 35sylan 487 . . . . . . . . . . 11 (((𝐹𝑠) ⊆ 𝑥 ∧ (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → (𝑜𝑥) ≠ ∅)
3736rexlimivw 3058 . . . . . . . . . 10 (∃𝑠𝐿 ((𝐹𝑠) ⊆ 𝑥 ∧ (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → (𝑜𝑥) ≠ ∅)
3833, 37syl 17 . . . . . . . . 9 ((∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥 ∧ ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → (𝑜𝑥) ≠ ∅)
3938ex 449 . . . . . . . 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 2998 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅ → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅))
4228, 41impbid 202 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅ ↔ ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
4342imbi2d 329 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → ((𝐴𝑜 → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅) ↔ (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅)))
4443ralbidva 3014 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (∀𝑜𝐽 (𝐴𝑜 → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅) ↔ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅)))
4544anbi2d 740 . 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 294 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  cin 3606  wss 3607  c0 3948  cima 5146  wf 5922  cfv 5926  (class class class)co 6690  fBascfbas 19782  filGencfg 19783  TopOnctopon 20763  Filcfil 21696   FilMap cfm 21784   fClus cfcls 21787   fClusf cfcf 21788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-fbas 19791  df-fg 19792  df-top 20747  df-topon 20764  df-cld 20871  df-ntr 20872  df-cls 20873  df-fil 21697  df-fm 21789  df-fcls 21792  df-fcf 21793
This theorem is referenced by:  fcfnei  21886
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