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Theorem flfssfcf 23763
Description: A limit point of a function is a cluster point of the function. (Contributed by Jeff Hankins, 28-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
Assertion
Ref Expression
flfssfcf ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ ((𝐽 fLimf 𝐿)β€˜πΉ) βŠ† ((𝐽 fClusf 𝐿)β€˜πΉ))
Proof of Theorem flfssfcf
StepHypRef Expression
1 flimfcls 23751 . . 3 (𝐽 fLim ((𝑋 FilMap 𝐹)β€˜πΏ)) βŠ† (𝐽 fClus ((𝑋 FilMap 𝐹)β€˜πΏ))
21a1i 11 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ (𝐽 fLim ((𝑋 FilMap 𝐹)β€˜πΏ)) βŠ† (𝐽 fClus ((𝑋 FilMap 𝐹)β€˜πΏ)))
3 flfval 23715 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ ((𝐽 fLimf 𝐿)β€˜πΉ) = (𝐽 fLim ((𝑋 FilMap 𝐹)β€˜πΏ)))
4 fcfval 23758 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ ((𝐽 fClusf 𝐿)β€˜πΉ) = (𝐽 fClus ((𝑋 FilMap 𝐹)β€˜πΏ)))
52, 3, 43sstr4d 4029 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ ((𝐽 fLimf 𝐿)β€˜πΉ) βŠ† ((𝐽 fClusf 𝐿)β€˜πΉ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1086   ∈ wcel 2105   βŠ† wss 3948  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7412  TopOnctopon 22633  Filcfil 23570   FilMap cfm 23658   fLim cflim 23659   fLimf cflf 23660   fClus cfcls 23661   fClusf cfcf 23662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8826  df-fbas 21142  df-top 22617  df-topon 22634  df-cld 22744  df-ntr 22745  df-cls 22746  df-nei 22823  df-fil 23571  df-flim 23664  df-flf 23665  df-fcls 23666  df-fcf 23667
This theorem is referenced by:  cnpfcfi  23765
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