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Theorem fcfelbas 23539
Description: A cluster point of a function is in the base set of the topology. (Contributed by Jeff Hankins, 26-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
Assertion
Ref Expression
fcfelbas (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝐴 ∈ ((𝐽 fClusf 𝐿)β€˜πΉ)) β†’ 𝐴 ∈ 𝑋)
Proof of Theorem fcfelbas
StepHypRef Expression
1 fcfval 23536 . . . . 5 ((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)))
21eleq2d 2819 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ (𝐴 ∈ ((𝐽 fClusf 𝐿)β€˜πΉ) ↔ 𝐴 ∈ (𝐽 fClus ((𝑋 FilMap 𝐹)β€˜πΏ))))
3 eqid 2732 . . . . 5 βˆͺ 𝐽 = βˆͺ 𝐽
43fclselbas 23519 . . . 4 (𝐴 ∈ (𝐽 fClus ((𝑋 FilMap 𝐹)β€˜πΏ)) β†’ 𝐴 ∈ βˆͺ 𝐽)
52, 4syl6bi 252 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ (𝐴 ∈ ((𝐽 fClusf 𝐿)β€˜πΉ) β†’ 𝐴 ∈ βˆͺ 𝐽))
65imp 407 . 2 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝐴 ∈ ((𝐽 fClusf 𝐿)β€˜πΉ)) β†’ 𝐴 ∈ βˆͺ 𝐽)
7 simpl1 1191 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝐴 ∈ ((𝐽 fClusf 𝐿)β€˜πΉ)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
8 toponuni 22415 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
97, 8syl 17 . 2 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝐴 ∈ ((𝐽 fClusf 𝐿)β€˜πΉ)) β†’ 𝑋 = βˆͺ 𝐽)
106, 9eleqtrrd 2836 1 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝐴 ∈ ((𝐽 fClusf 𝐿)β€˜πΉ)) β†’ 𝐴 ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆͺ cuni 4908  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408  TopOnctopon 22411  Filcfil 23348   FilMap cfm 23436   fClus cfcls 23439   fClusf cfcf 23440
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
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 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 7411  df-oprab 7412  df-mpo 7413  df-map 8821  df-fbas 20940  df-top 22395  df-topon 22412  df-cld 22522  df-ntr 22523  df-cls 22524  df-fil 23349  df-fcls 23444  df-fcf 23445
This theorem is referenced by: (None)
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