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Theorem flfval 21995
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 20932 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
2 filtop 21860 . . . . 5 (𝐿 ∈ (Fil‘𝑌) → 𝑌𝐿)
3 elmapg 8036 . . . . 5 ((𝑋𝐽𝑌𝐿) → (𝐹 ∈ (𝑋𝑚 𝑌) ↔ 𝐹:𝑌𝑋))
41, 2, 3syl2an 495 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐹 ∈ (𝑋𝑚 𝑌) ↔ 𝐹:𝑌𝑋))
54biimpar 503 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ 𝐹:𝑌𝑋) → 𝐹 ∈ (𝑋𝑚 𝑌))
6 flffval 21994 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLimf 𝐿) = (𝑓 ∈ (𝑋𝑚 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿))))
76fveq1d 6354 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → ((𝐽 fLimf 𝐿)‘𝐹) = ((𝑓 ∈ (𝑋𝑚 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))‘𝐹))
8 oveq2 6821 . . . . . . 7 (𝑓 = 𝐹 → (𝑋 FilMap 𝑓) = (𝑋 FilMap 𝐹))
98fveq1d 6354 . . . . . 6 (𝑓 = 𝐹 → ((𝑋 FilMap 𝑓)‘𝐿) = ((𝑋 FilMap 𝐹)‘𝐿))
109oveq2d 6829 . . . . 5 (𝑓 = 𝐹 → (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
11 eqid 2760 . . . . 5 (𝑓 ∈ (𝑋𝑚 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿))) = (𝑓 ∈ (𝑋𝑚 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))
12 ovex 6841 . . . . 5 (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)) ∈ V
1310, 11, 12fvmpt 6444 . . . 4 (𝐹 ∈ (𝑋𝑚 𝑌) → ((𝑓 ∈ (𝑋𝑚 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
147, 13sylan9eq 2814 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ 𝐹 ∈ (𝑋𝑚 𝑌)) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
155, 14syldan 488 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ 𝐹:𝑌𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
16153impa 1101 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  cmpt 4881  wf 6045  cfv 6049  (class class class)co 6813  𝑚 cmap 8023  TopOnctopon 20917  Filcfil 21850   FilMap cfm 21938   fLim cflim 21939   fLimf cflf 21940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-map 8025  df-fbas 19945  df-top 20901  df-topon 20918  df-fil 21851  df-flf 21945
This theorem is referenced by:  flfnei  21996  isflf  21998  hausflf  22002  flfcnp  22009  flfssfcf  22043  uffcfflf  22044  cnpfcf  22046  cnextcn  22072  tsmscls  22142  cnextucn  22308  cmetcaulem  23286  fmcncfil  30286
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