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Theorem fcfval 23537
Description: The set of cluster points of a function. (Contributed by Jeff Hankins, 24-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
Assertion
Ref Expression
fcfval ((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)))
Proof of Theorem fcfval
Dummy variables 𝑓 𝑔 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fcf 23446 . . . . 5 fClusf = (𝑗 ∈ Top, 𝑓 ∈ βˆͺ ran Fil ↦ (𝑔 ∈ (βˆͺ 𝑗 ↑m βˆͺ 𝑓) ↦ (𝑗 fClus ((βˆͺ 𝑗 FilMap 𝑔)β€˜π‘“))))
21a1i 11 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ fClusf = (𝑗 ∈ Top, 𝑓 ∈ βˆͺ ran Fil ↦ (𝑔 ∈ (βˆͺ 𝑗 ↑m βˆͺ 𝑓) ↦ (𝑗 fClus ((βˆͺ 𝑗 FilMap 𝑔)β€˜π‘“)))))
3 simprl 770 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) β†’ 𝑗 = 𝐽)
43unieqd 4923 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
5 toponuni 22416 . . . . . . . 8 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
65ad2antrr 725 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) β†’ 𝑋 = βˆͺ 𝐽)
74, 6eqtr4d 2776 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) β†’ βˆͺ 𝑗 = 𝑋)
8 unieq 4920 . . . . . . . 8 (𝑓 = 𝐿 β†’ βˆͺ 𝑓 = βˆͺ 𝐿)
98ad2antll 728 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) β†’ βˆͺ 𝑓 = βˆͺ 𝐿)
10 filunibas 23385 . . . . . . . 8 (𝐿 ∈ (Filβ€˜π‘Œ) β†’ βˆͺ 𝐿 = π‘Œ)
1110ad2antlr 726 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) β†’ βˆͺ 𝐿 = π‘Œ)
129, 11eqtrd 2773 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) β†’ βˆͺ 𝑓 = π‘Œ)
137, 12oveq12d 7427 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) β†’ (βˆͺ 𝑗 ↑m βˆͺ 𝑓) = (𝑋 ↑m π‘Œ))
147oveq1d 7424 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) β†’ (βˆͺ 𝑗 FilMap 𝑔) = (𝑋 FilMap 𝑔))
15 simprr 772 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) β†’ 𝑓 = 𝐿)
1614, 15fveq12d 6899 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) β†’ ((βˆͺ 𝑗 FilMap 𝑔)β€˜π‘“) = ((𝑋 FilMap 𝑔)β€˜πΏ))
173, 16oveq12d 7427 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) β†’ (𝑗 fClus ((βˆͺ 𝑗 FilMap 𝑔)β€˜π‘“)) = (𝐽 fClus ((𝑋 FilMap 𝑔)β€˜πΏ)))
1813, 17mpteq12dv 5240 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) β†’ (𝑔 ∈ (βˆͺ 𝑗 ↑m βˆͺ 𝑓) ↦ (𝑗 fClus ((βˆͺ 𝑗 FilMap 𝑔)β€˜π‘“))) = (𝑔 ∈ (𝑋 ↑m π‘Œ) ↦ (𝐽 fClus ((𝑋 FilMap 𝑔)β€˜πΏ))))
19 topontop 22415 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2019adantr 482 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ 𝐽 ∈ Top)
21 fvssunirn 6925 . . . . . 6 (Filβ€˜π‘Œ) βŠ† βˆͺ ran Fil
2221sseli 3979 . . . . 5 (𝐿 ∈ (Filβ€˜π‘Œ) β†’ 𝐿 ∈ βˆͺ ran Fil)
2322adantl 483 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ 𝐿 ∈ βˆͺ ran Fil)
24 ovex 7442 . . . . . 6 (𝑋 ↑m π‘Œ) ∈ V
2524mptex 7225 . . . . 5 (𝑔 ∈ (𝑋 ↑m π‘Œ) ↦ (𝐽 fClus ((𝑋 FilMap 𝑔)β€˜πΏ))) ∈ V
2625a1i 11 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ (𝑔 ∈ (𝑋 ↑m π‘Œ) ↦ (𝐽 fClus ((𝑋 FilMap 𝑔)β€˜πΏ))) ∈ V)
272, 18, 20, 23, 26ovmpod 7560 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ (𝐽 fClusf 𝐿) = (𝑔 ∈ (𝑋 ↑m π‘Œ) ↦ (𝐽 fClus ((𝑋 FilMap 𝑔)β€˜πΏ))))
28273adant3 1133 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ (𝐽 fClusf 𝐿) = (𝑔 ∈ (𝑋 ↑m π‘Œ) ↦ (𝐽 fClus ((𝑋 FilMap 𝑔)β€˜πΏ))))
29 simpr 486 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝑔 = 𝐹) β†’ 𝑔 = 𝐹)
3029oveq2d 7425 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝑔 = 𝐹) β†’ (𝑋 FilMap 𝑔) = (𝑋 FilMap 𝐹))
3130fveq1d 6894 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝑔 = 𝐹) β†’ ((𝑋 FilMap 𝑔)β€˜πΏ) = ((𝑋 FilMap 𝐹)β€˜πΏ))
3231oveq2d 7425 . 2 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝑔 = 𝐹) β†’ (𝐽 fClus ((𝑋 FilMap 𝑔)β€˜πΏ)) = (𝐽 fClus ((𝑋 FilMap 𝐹)β€˜πΏ)))
33 toponmax 22428 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
34 filtop 23359 . . . 4 (𝐿 ∈ (Filβ€˜π‘Œ) β†’ π‘Œ ∈ 𝐿)
35 elmapg 8833 . . . 4 ((𝑋 ∈ 𝐽 ∧ π‘Œ ∈ 𝐿) β†’ (𝐹 ∈ (𝑋 ↑m π‘Œ) ↔ 𝐹:π‘ŒβŸΆπ‘‹))
3633, 34, 35syl2an 597 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝑋 ↑m π‘Œ) ↔ 𝐹:π‘ŒβŸΆπ‘‹))
3736biimp3ar 1471 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ 𝐹 ∈ (𝑋 ↑m π‘Œ))
38 ovexd 7444 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ (𝐽 fClus ((𝑋 FilMap 𝐹)β€˜πΏ)) ∈ V)
3928, 32, 37, 38fvmptd 7006 1 ((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)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  Vcvv 3475  βˆͺ cuni 4909   ↦ cmpt 5232  ran crn 5678  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411   ↑m cmap 8820  Topctop 22395  TopOnctopon 22412  Filcfil 23349   FilMap cfm 23437   fClus cfcls 23440   fClusf cfcf 23441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-fbas 20941  df-top 22396  df-topon 22413  df-fil 23350  df-fcf 23446
This theorem is referenced by:  isfcf  23538  fcfelbas  23540  flfssfcf  23542  uffcfflf  23543  cnpfcfi  23544  cnpfcf  23545
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