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Theorem flfval 23485
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 22419 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
2 filtop 23350 . . . . 5 (𝐿 ∈ (Fil‘𝑌) → 𝑌𝐿)
3 elmapg 8829 . . . . 5 ((𝑋𝐽𝑌𝐿) → (𝐹 ∈ (𝑋m 𝑌) ↔ 𝐹:𝑌𝑋))
41, 2, 3syl2an 596 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐹 ∈ (𝑋m 𝑌) ↔ 𝐹:𝑌𝑋))
54biimpar 478 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ 𝐹:𝑌𝑋) → 𝐹 ∈ (𝑋m 𝑌))
6 flffval 23484 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLimf 𝐿) = (𝑓 ∈ (𝑋m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿))))
76fveq1d 6890 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → ((𝐽 fLimf 𝐿)‘𝐹) = ((𝑓 ∈ (𝑋m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))‘𝐹))
8 oveq2 7413 . . . . . . 7 (𝑓 = 𝐹 → (𝑋 FilMap 𝑓) = (𝑋 FilMap 𝐹))
98fveq1d 6890 . . . . . 6 (𝑓 = 𝐹 → ((𝑋 FilMap 𝑓)‘𝐿) = ((𝑋 FilMap 𝐹)‘𝐿))
109oveq2d 7421 . . . . 5 (𝑓 = 𝐹 → (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
11 eqid 2732 . . . . 5 (𝑓 ∈ (𝑋m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿))) = (𝑓 ∈ (𝑋m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))
12 ovex 7438 . . . . 5 (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)) ∈ V
1310, 11, 12fvmpt 6995 . . . 4 (𝐹 ∈ (𝑋m 𝑌) → ((𝑓 ∈ (𝑋m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
147, 13sylan9eq 2792 . . 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  cmpt 5230  wf 6536  cfv 6540  (class class class)co 7405  m cmap 8816  TopOnctopon 22403  Filcfil 23340   FilMap cfm 23428   fLim cflim 23429   fLimf cflf 23430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8818  df-fbas 20933  df-top 22387  df-topon 22404  df-fil 23341  df-flf 23435
This theorem is referenced by:  flfnei  23486  isflf  23488  hausflf  23492  flfcnp  23499  flfssfcf  23533  uffcfflf  23534  cnpfcf  23536  cnextcn  23562  tsmscls  23633  cnextucn  23799  cmetcaulem  24796  fmcncfil  32899
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