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Theorem fcfval 23528
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 23437 . . . . 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 769 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) β†’ 𝑗 = 𝐽)
43unieqd 4921 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) β†’ βˆͺ 𝑗 = βˆͺ 𝐽)
5 toponuni 22407 . . . . . . . 8 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
65ad2antrr 724 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) β†’ 𝑋 = βˆͺ 𝐽)
74, 6eqtr4d 2775 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) β†’ βˆͺ 𝑗 = 𝑋)
8 unieq 4918 . . . . . . . 8 (𝑓 = 𝐿 β†’ βˆͺ 𝑓 = βˆͺ 𝐿)
98ad2antll 727 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) β†’ βˆͺ 𝑓 = βˆͺ 𝐿)
10 filunibas 23376 . . . . . . . 8 (𝐿 ∈ (Filβ€˜π‘Œ) β†’ βˆͺ 𝐿 = π‘Œ)
1110ad2antlr 725 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) β†’ βˆͺ 𝐿 = π‘Œ)
129, 11eqtrd 2772 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) β†’ βˆͺ 𝑓 = π‘Œ)
137, 12oveq12d 7423 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) β†’ (βˆͺ 𝑗 ↑m βˆͺ 𝑓) = (𝑋 ↑m π‘Œ))
147oveq1d 7420 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) β†’ (βˆͺ 𝑗 FilMap 𝑔) = (𝑋 FilMap 𝑔))
15 simprr 771 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) β†’ 𝑓 = 𝐿)
1614, 15fveq12d 6895 . . . . . 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 7423 . . . . 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 5238 . . . 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 22406 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2019adantr 481 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ 𝐽 ∈ Top)
21 fvssunirn 6921 . . . . . 6 (Filβ€˜π‘Œ) βŠ† βˆͺ ran Fil
2221sseli 3977 . . . . 5 (𝐿 ∈ (Filβ€˜π‘Œ) β†’ 𝐿 ∈ βˆͺ ran Fil)
2322adantl 482 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ 𝐿 ∈ βˆͺ ran Fil)
24 ovex 7438 . . . . . 6 (𝑋 ↑m π‘Œ) ∈ V
2524mptex 7221 . . . . 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 7556 . . 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 1132 . 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 485 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝑔 = 𝐹) β†’ 𝑔 = 𝐹)
3029oveq2d 7421 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) ∧ 𝑔 = 𝐹) β†’ (𝑋 FilMap 𝑔) = (𝑋 FilMap 𝐹))
3130fveq1d 6890 . . 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 7421 . 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 22419 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
34 filtop 23350 . . . 4 (𝐿 ∈ (Filβ€˜π‘Œ) β†’ π‘Œ ∈ 𝐿)
35 elmapg 8829 . . . 4 ((𝑋 ∈ 𝐽 ∧ π‘Œ ∈ 𝐿) β†’ (𝐹 ∈ (𝑋 ↑m π‘Œ) ↔ 𝐹:π‘ŒβŸΆπ‘‹))
3633, 34, 35syl2an 596 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝑋 ↑m π‘Œ) ↔ 𝐹:π‘ŒβŸΆπ‘‹))
3736biimp3ar 1470 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ 𝐹 ∈ (𝑋 ↑m π‘Œ))
38 ovexd 7440 . 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 7002 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3474  βˆͺ cuni 4907   ↦ cmpt 5230  ran crn 5676  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407   ↑m cmap 8816  Topctop 22386  TopOnctopon 22403  Filcfil 23340   FilMap cfm 23428   fClus cfcls 23431   fClusf cfcf 23432
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-fcf 23437
This theorem is referenced by:  isfcf  23529  fcfelbas  23531  flfssfcf  23533  uffcfflf  23534  cnpfcfi  23535  cnpfcf  23536
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