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Theorem isfcf 22166
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 22165 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐽 fClusf 𝐿)‘𝐹) = (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)))
21eleq2d 2864 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ 𝐴 ∈ (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿))))
3 simp1 1167 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝐽 ∈ (TopOn‘𝑋))
4 toponmax 21059 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
5 filfbas 21980 . . . 4 (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
6 id 22 . . . 4 (𝐹:𝑌𝑋𝐹:𝑌𝑋)
7 fmfil 22076 . . . 4 ((𝑋𝐽𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
84, 5, 6, 7syl3an 1200 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
9 fclsopn 22146 . . 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 580 . 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 1270 . . . . . . . . . 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 1272 . . . . . . . . . 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 1274 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) ∧ 𝑠𝐿) → 𝐹:𝑌𝑋)
16 simpl2 1245 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → 𝐿 ∈ (Fil‘𝑌))
17 fgfil 22007 . . . . . . . . . . . 12 (𝐿 ∈ (Fil‘𝑌) → (𝑌filGen𝐿) = 𝐿)
1816, 17syl 17 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (𝑌filGen𝐿) = 𝐿)
1918eleq2d 2864 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (𝑠 ∈ (𝑌filGen𝐿) ↔ 𝑠𝐿))
2019biimpar 470 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) ∧ 𝑠𝐿) → 𝑠 ∈ (𝑌filGen𝐿))
21 eqid 2799 . . . . . . . . . 10 (𝑌filGen𝐿) = (𝑌filGen𝐿)
2221imaelfm 22083 . . . . . . . . 9 (((𝑋𝐽𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑠 ∈ (𝑌filGen𝐿)) → (𝐹𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐿))
2312, 14, 15, 20, 22syl31anc 1493 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) ∧ 𝑠𝐿) → (𝐹𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐿))
24 ineq2 4006 . . . . . . . . . 10 (𝑥 = (𝐹𝑠) → (𝑜𝑥) = (𝑜 ∩ (𝐹𝑠)))
2524neeq1d 3030 . . . . . . . . 9 (𝑥 = (𝐹𝑠) → ((𝑜𝑥) ≠ ∅ ↔ (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
2625rspcv 3493 . . . . . . . 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 3150 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅ → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
29 elfm 22079 . . . . . . . . . . 11 ((𝑋𝐽𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑥𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥)))
304, 5, 6, 29syl3an 1200 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑥𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥)))
3130adantr 473 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑥𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥)))
3231simplbda 494 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) ∧ 𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)) → ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥)
33 r19.29r 3254 . . . . . . . . . 10 ((∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥 ∧ ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → ∃𝑠𝐿 ((𝐹𝑠) ⊆ 𝑥 ∧ (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
34 sslin 4034 . . . . . . . . . . . 12 ((𝐹𝑠) ⊆ 𝑥 → (𝑜 ∩ (𝐹𝑠)) ⊆ (𝑜𝑥))
35 ssn0 4172 . . . . . . . . . . . 12 (((𝑜 ∩ (𝐹𝑠)) ⊆ (𝑜𝑥) ∧ (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → (𝑜𝑥) ≠ ∅)
3634, 35sylan 576 . . . . . . . . . . 11 (((𝐹𝑠) ⊆ 𝑥 ∧ (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → (𝑜𝑥) ≠ ∅)
3736rexlimivw 3210 . . . . . . . . . 10 (∃𝑠𝐿 ((𝐹𝑠) ⊆ 𝑥 ∧ (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → (𝑜𝑥) ≠ ∅)
3833, 37syl 17 . . . . . . . . 9 ((∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥 ∧ ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → (𝑜𝑥) ≠ ∅)
3938ex 402 . . . . . . . 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 3150 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅ → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅))
4228, 41impbid 204 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅ ↔ ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
4342imbi2d 332 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → ((𝐴𝑜 → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅) ↔ (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅)))
4443ralbidva 3166 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (∀𝑜𝐽 (𝐴𝑜 → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅) ↔ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅)))
4544anbi2d 623 . 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 297 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 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wne 2971  wral 3089  wrex 3090  cin 3768  wss 3769  c0 4115  cima 5315  wf 6097  cfv 6101  (class class class)co 6878  fBascfbas 20056  filGencfg 20057  TopOnctopon 21043  Filcfil 21977   FilMap cfm 22065   fClus cfcls 22068   fClusf cfcf 22069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-iin 4713  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-map 8097  df-fbas 20065  df-fg 20066  df-top 21027  df-topon 21044  df-cld 21152  df-ntr 21153  df-cls 21154  df-fil 21978  df-fm 22070  df-fcls 22073  df-fcf 22074
This theorem is referenced by:  fcfnei  22167
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