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

Theorem isfcf 22326
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 22325 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐽 fClusf 𝐿)‘𝐹) = (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)))
21eleq2d 2868 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ 𝐴 ∈ (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿))))
3 simp1 1129 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝐽 ∈ (TopOn‘𝑋))
4 toponmax 21218 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
5 filfbas 22140 . . . 4 (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
6 id 22 . . . 4 (𝐹:𝑌𝑋𝐹:𝑌𝑋)
7 fmfil 22236 . . . 4 ((𝑋𝐽𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
84, 5, 6, 7syl3an 1153 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋))
9 fclsopn 22306 . . 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 1205 . . . . . . . . . 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 1206 . . . . . . . . . 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 1207 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) ∧ 𝑠𝐿) → 𝐹:𝑌𝑋)
16 simpl2 1185 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → 𝐿 ∈ (Fil‘𝑌))
17 fgfil 22167 . . . . . . . . . . . 12 (𝐿 ∈ (Fil‘𝑌) → (𝑌filGen𝐿) = 𝐿)
1816, 17syl 17 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (𝑌filGen𝐿) = 𝐿)
1918eleq2d 2868 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (𝑠 ∈ (𝑌filGen𝐿) ↔ 𝑠𝐿))
2019biimpar 478 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) ∧ 𝑠𝐿) → 𝑠 ∈ (𝑌filGen𝐿))
21 eqid 2795 . . . . . . . . . 10 (𝑌filGen𝐿) = (𝑌filGen𝐿)
2221imaelfm 22243 . . . . . . . . 9 (((𝑋𝐽𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑠 ∈ (𝑌filGen𝐿)) → (𝐹𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐿))
2312, 14, 15, 20, 22syl31anc 1366 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) ∧ 𝑠𝐿) → (𝐹𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐿))
24 ineq2 4103 . . . . . . . . . 10 (𝑥 = (𝐹𝑠) → (𝑜𝑥) = (𝑜 ∩ (𝐹𝑠)))
2524neeq1d 3043 . . . . . . . . 9 (𝑥 = (𝐹𝑠) → ((𝑜𝑥) ≠ ∅ ↔ (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
2625rspcv 3555 . . . . . . . 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 3156 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅ → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
29 elfm 22239 . . . . . . . . . . 11 ((𝑋𝐽𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑥𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥)))
304, 5, 6, 29syl3an 1153 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑥𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥)))
3130adantr 481 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑥𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥)))
3231simplbda 500 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) ∧ 𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)) → ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥)
33 r19.29r 3219 . . . . . . . . . 10 ((∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥 ∧ ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → ∃𝑠𝐿 ((𝐹𝑠) ⊆ 𝑥 ∧ (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
34 sslin 4131 . . . . . . . . . . . 12 ((𝐹𝑠) ⊆ 𝑥 → (𝑜 ∩ (𝐹𝑠)) ⊆ (𝑜𝑥))
35 ssn0 4274 . . . . . . . . . . . 12 (((𝑜 ∩ (𝐹𝑠)) ⊆ (𝑜𝑥) ∧ (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → (𝑜𝑥) ≠ ∅)
3634, 35sylan 580 . . . . . . . . . . 11 (((𝐹𝑠) ⊆ 𝑥 ∧ (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → (𝑜𝑥) ≠ ∅)
3736rexlimivw 3245 . . . . . . . . . 10 (∃𝑠𝐿 ((𝐹𝑠) ⊆ 𝑥 ∧ (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → (𝑜𝑥) ≠ ∅)
3833, 37syl 17 . . . . . . . . 9 ((∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥 ∧ ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅) → (𝑜𝑥) ≠ ∅)
3938ex 413 . . . . . . . 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 3156 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅ → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅))
4228, 41impbid 213 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → (∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅ ↔ ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅))
4342imbi2d 342 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑜𝐽) → ((𝐴𝑜 → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅) ↔ (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅)))
4443ralbidva 3163 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (∀𝑜𝐽 (𝐴𝑜 → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜𝑥) ≠ ∅) ↔ ∀𝑜𝐽 (𝐴𝑜 → ∀𝑠𝐿 (𝑜 ∩ (𝐹𝑠)) ≠ ∅)))
4544anbi2d 628 . 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 306 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 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wne 2984  wral 3105  wrex 3106  cin 3858  wss 3859  c0 4211  cima 5446  wf 6221  cfv 6225  (class class class)co 7016  fBascfbas 20215  filGencfg 20216  TopOnctopon 21202  Filcfil 22137   FilMap cfm 22225   fClus cfcls 22228   fClusf cfcf 22229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-iin 4828  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-map 8258  df-fbas 20224  df-fg 20225  df-top 21186  df-topon 21203  df-cld 21311  df-ntr 21312  df-cls 21313  df-fil 22138  df-fm 22230  df-fcls 22233  df-fcf 22234
This theorem is referenced by:  fcfnei  22327
  Copyright terms: Public domain W3C validator