MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isfcf Structured version   Visualization version   GIF version

Theorem isfcf 24042
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 24041 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐽 fClusf 𝐿)‘𝐹) = (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)))
21eleq2d 2827 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ 𝐴 ∈ (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿))))
3 simp1 1137 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝐽 ∈ (TopOn‘𝑋))
4 toponmax 22932 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
5 filfbas 23856 . . . 4 (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
6 id 22 . . . 4 (𝐹:𝑌𝑋𝐹:𝑌𝑋)
7 fmfil 23952 . . . 4 ((𝑋𝐽𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
84, 5, 6, 7syl3an 1161 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
9 fclsopn 24022 . . 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 1213 . . . . . . . . . 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 1214 . . . . . . . . . 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 1215 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) ∧ 𝑠𝐿) → 𝐹:𝑌𝑋)
16 simpl2 1193 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → 𝐿 ∈ (Fil‘𝑌))
17 fgfil 23883 . . . . . . . . . . . 12 (𝐿 ∈ (Fil‘𝑌) → (𝑌filGen𝐿) = 𝐿)
1816, 17syl 17 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (𝑌filGen𝐿) = 𝐿)
1918eleq2d 2827 . . . . . . . . . 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 2737 . . . . . . . . . 10 (𝑌filGen𝐿) = (𝑌filGen𝐿)
2221imaelfm 23959 . . . . . . . . 9 (((𝑋𝐽𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑠 ∈ (𝑌filGen𝐿)) → (𝐹𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐿))
2312, 14, 15, 20, 22syl31anc 1375 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) ∧ 𝑠𝐿) → (𝐹𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐿))
24 ineq2 4214 . . . . . . . . . 10 (𝑥 = (𝐹𝑠) → (𝑜𝑥) = (𝑜 ∩ (𝐹𝑠)))
2524neeq1d 3000 . . . . . . . . 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 3154 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅ → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
29 elfm 23955 . . . . . . . . . . 11 ((𝑋𝐽𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑥𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥)))
304, 5, 6, 29syl3an 1161 . . . . . . . . . 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 3116 . . . . . . . . . 10 ((∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥 ∧ ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → ∃𝑠𝐿 ((𝐹𝑠) ⊆ 𝑥 ∧ (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
34 sslin 4243 . . . . . . . . . . . 12 ((𝐹𝑠) ⊆ 𝑥 → (𝑜 ∩ (𝐹𝑠)) ⊆ (𝑜𝑥))
35 ssn0 4404 . . . . . . . . . . . 12 (((𝑜 ∩ (𝐹𝑠)) ⊆ (𝑜𝑥) ∧ (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → (𝑜𝑥) ≠ ∅)
3634, 35sylan 580 . . . . . . . . . . 11 (((𝐹𝑠) ⊆ 𝑥 ∧ (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → (𝑜𝑥) ≠ ∅)
3736rexlimivw 3151 . . . . . . . . . 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 3154 . . . . . 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 3176 . . 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 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  cin 3950  wss 3951  c0 4333  cima 5688  wf 6557  cfv 6561  (class class class)co 7431  fBascfbas 21352  filGencfg 21353  TopOnctopon 22916  Filcfil 23853   FilMap cfm 23941   fClus cfcls 23944   fClusf cfcf 23945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-fbas 21361  df-fg 21362  df-top 22900  df-topon 22917  df-cld 23027  df-ntr 23028  df-cls 23029  df-fil 23854  df-fm 23946  df-fcls 23949  df-fcf 23950
This theorem is referenced by:  fcfnei  24043
  Copyright terms: Public domain W3C validator