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Theorem flfval 23494
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 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ ((𝐽 fLimf 𝐿)β€˜πΉ) = (𝐽 fLim ((𝑋 FilMap 𝐹)β€˜πΏ)))
Proof of Theorem flfval
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 toponmax 22428 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
2 filtop 23359 . . . . 5 (𝐿 ∈ (Filβ€˜π‘Œ) β†’ π‘Œ ∈ 𝐿)
3 elmapg 8833 . . . . 5 ((𝑋 ∈ 𝐽 ∧ π‘Œ ∈ 𝐿) β†’ (𝐹 ∈ (𝑋 ↑m π‘Œ) ↔ 𝐹:π‘ŒβŸΆπ‘‹))
41, 2, 3syl2an 597 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝑋 ↑m π‘Œ) ↔ 𝐹:π‘ŒβŸΆπ‘‹))
54biimpar 479 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ 𝐹 ∈ (𝑋 ↑m π‘Œ))
6 flffval 23493 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ (𝐽 fLimf 𝐿) = (𝑓 ∈ (𝑋 ↑m π‘Œ) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)β€˜πΏ))))
76fveq1d 6894 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ ((𝐽 fLimf 𝐿)β€˜πΉ) = ((𝑓 ∈ (𝑋 ↑m π‘Œ) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)β€˜πΏ)))β€˜πΉ))
8 oveq2 7417 . . . . . . 7 (𝑓 = 𝐹 β†’ (𝑋 FilMap 𝑓) = (𝑋 FilMap 𝐹))
98fveq1d 6894 . . . . . 6 (𝑓 = 𝐹 β†’ ((𝑋 FilMap 𝑓)β€˜πΏ) = ((𝑋 FilMap 𝐹)β€˜πΏ))
109oveq2d 7425 . . . . 5 (𝑓 = 𝐹 β†’ (𝐽 fLim ((𝑋 FilMap 𝑓)β€˜πΏ)) = (𝐽 fLim ((𝑋 FilMap 𝐹)β€˜πΏ)))
11 eqid 2733 . . . . 5 (𝑓 ∈ (𝑋 ↑m π‘Œ) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)β€˜πΏ))) = (𝑓 ∈ (𝑋 ↑m π‘Œ) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)β€˜πΏ)))
12 ovex 7442 . . . . 5 (𝐽 fLim ((𝑋 FilMap 𝐹)β€˜πΏ)) ∈ V
1310, 11, 12fvmpt 6999 . . . 4 (𝐹 ∈ (𝑋 ↑m π‘Œ) β†’ ((𝑓 ∈ (𝑋 ↑m π‘Œ) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)β€˜πΏ)))β€˜πΉ) = (𝐽 fLim ((𝑋 FilMap 𝐹)β€˜πΏ)))
147, 13sylan9eq 2793 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ 𝐹 ∈ (𝑋 ↑m π‘Œ)) β†’ ((𝐽 fLimf 𝐿)β€˜πΉ) = (𝐽 fLim ((𝑋 FilMap 𝐹)β€˜πΏ)))
155, 14syldan 592 . 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)))
16153impa 1111 1 ((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)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ↦ cmpt 5232  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ↑m cmap 8820  TopOnctopon 22412  Filcfil 23349   FilMap cfm 23437   fLim cflim 23438   fLimf cflf 23439
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-flf 23444
This theorem is referenced by:  flfnei  23495  isflf  23497  hausflf  23501  flfcnp  23508  flfssfcf  23542  uffcfflf  23543  cnpfcf  23545  cnextcn  23571  tsmscls  23642  cnextucn  23808  cmetcaulem  24805  fmcncfil  32942
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