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

Theorem flfval 23998
Description: Given a function from a filtered set to a topological space, define the set of limit points of the function. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Assertion
Ref Expression
flfval ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))

Proof of Theorem flfval
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 toponmax 22932 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
2 filtop 23863 . . . . 5 (𝐿 ∈ (Fil‘𝑌) → 𝑌𝐿)
3 elmapg 8879 . . . . 5 ((𝑋𝐽𝑌𝐿) → (𝐹 ∈ (𝑋m 𝑌) ↔ 𝐹:𝑌𝑋))
41, 2, 3syl2an 596 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐹 ∈ (𝑋m 𝑌) ↔ 𝐹:𝑌𝑋))
54biimpar 477 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ 𝐹:𝑌𝑋) → 𝐹 ∈ (𝑋m 𝑌))
6 flffval 23997 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLimf 𝐿) = (𝑓 ∈ (𝑋m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿))))
76fveq1d 6908 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → ((𝐽 fLimf 𝐿)‘𝐹) = ((𝑓 ∈ (𝑋m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))‘𝐹))
8 oveq2 7439 . . . . . . 7 (𝑓 = 𝐹 → (𝑋 FilMap 𝑓) = (𝑋 FilMap 𝐹))
98fveq1d 6908 . . . . . 6 (𝑓 = 𝐹 → ((𝑋 FilMap 𝑓)‘𝐿) = ((𝑋 FilMap 𝐹)‘𝐿))
109oveq2d 7447 . . . . 5 (𝑓 = 𝐹 → (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
11 eqid 2737 . . . . 5 (𝑓 ∈ (𝑋m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿))) = (𝑓 ∈ (𝑋m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))
12 ovex 7464 . . . . 5 (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)) ∈ V
1310, 11, 12fvmpt 7016 . . . 4 (𝐹 ∈ (𝑋m 𝑌) → ((𝑓 ∈ (𝑋m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
147, 13sylan9eq 2797 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ 𝐹 ∈ (𝑋m 𝑌)) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
155, 14syldan 591 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ 𝐹:𝑌𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
16153impa 1110 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  cmpt 5225  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  TopOnctopon 22916  Filcfil 23853   FilMap cfm 23941   fLim cflim 23942   fLimf cflf 23943
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-iun 4993  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-top 22900  df-topon 22917  df-fil 23854  df-flf 23948
This theorem is referenced by:  flfnei  23999  isflf  24001  hausflf  24005  flfcnp  24012  flfssfcf  24046  uffcfflf  24047  cnpfcf  24049  cnextcn  24075  tsmscls  24146  cnextucn  24312  cmetcaulem  25322  fmcncfil  33930
  Copyright terms: Public domain W3C validator