MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  flfval Structured version   Visualization version   GIF version
Theorem flfval 23716
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 22650 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
2 filtop 23581 . . . . 5 (𝐿 ∈ (Filβ€˜π‘Œ) β†’ π‘Œ ∈ 𝐿)
3 elmapg 8837 . . . . 5 ((𝑋 ∈ 𝐽 ∧ π‘Œ ∈ 𝐿) β†’ (𝐹 ∈ (𝑋 ↑m π‘Œ) ↔ 𝐹:π‘ŒβŸΆπ‘‹))
41, 2, 3syl2an 594 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝑋 ↑m π‘Œ) ↔ 𝐹:π‘ŒβŸΆπ‘‹))
54biimpar 476 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) ∧ 𝐹:π‘ŒβŸΆπ‘‹) β†’ 𝐹 ∈ (𝑋 ↑m π‘Œ))
6 flffval 23715 . . . . 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 7421 . . . . . . 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 7429 . . . . 5 (𝑓 = 𝐹 β†’ (𝐽 fLim ((𝑋 FilMap 𝑓)β€˜πΏ)) = (𝐽 fLim ((𝑋 FilMap 𝐹)β€˜πΏ)))
11 eqid 2730 . . . . 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 7446 . . . . 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 2790 . . 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 589 . 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 1108 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   ↦ cmpt 5232  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7413   ↑m cmap 8824  TopOnctopon 22634  Filcfil 23571   FilMap cfm 23659   fLim cflim 23660   fLimf cflf 23661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  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 7416  df-oprab 7417  df-mpo 7418  df-map 8826  df-fbas 21143  df-top 22618  df-topon 22635  df-fil 23572  df-flf 23666
This theorem is referenced by:  flfnei  23717  isflf  23719  hausflf  23723  flfcnp  23730  flfssfcf  23764  uffcfflf  23765  cnpfcf  23767  cnextcn  23793  tsmscls  23864  cnextucn  24030  cmetcaulem  25038  fmcncfil  33207
  Copyright terms: Public domain W3C validator