MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  flffval Structured version   Visualization version   GIF version
Theorem flffval 23880
Description: Given a topology and a filtered set, return the convergence function on the functions from the filtered set to the base set of the topological space. (Contributed by Jeff Hankins, 14-Oct-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Assertion
Ref Expression
flffval ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ (𝐽 fLimf 𝐿) = (𝑓 ∈ (𝑋 ↑m π‘Œ) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)β€˜πΏ))))
Distinct variable groups:   𝑓,𝐽   𝑓,𝑋   𝑓,π‘Œ   𝑓,𝐿
Proof of Theorem flffval
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 topontop 22802 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2 fvssunirn 6924 . . . 4 (Filβ€˜π‘Œ) βŠ† βˆͺ ran Fil
32sseli 3974 . . 3 (𝐿 ∈ (Filβ€˜π‘Œ) β†’ 𝐿 ∈ βˆͺ ran Fil)
4 unieq 4914 . . . . . 6 (π‘₯ = 𝐽 β†’ βˆͺ π‘₯ = βˆͺ 𝐽)
5 unieq 4914 . . . . . 6 (𝑦 = 𝐿 β†’ βˆͺ 𝑦 = βˆͺ 𝐿)
64, 5oveqan12d 7433 . . . . 5 ((π‘₯ = 𝐽 ∧ 𝑦 = 𝐿) β†’ (βˆͺ π‘₯ ↑m βˆͺ 𝑦) = (βˆͺ 𝐽 ↑m βˆͺ 𝐿))
7 simpl 482 . . . . . 6 ((π‘₯ = 𝐽 ∧ 𝑦 = 𝐿) β†’ π‘₯ = 𝐽)
84adantr 480 . . . . . . . 8 ((π‘₯ = 𝐽 ∧ 𝑦 = 𝐿) β†’ βˆͺ π‘₯ = βˆͺ 𝐽)
98oveq1d 7429 . . . . . . 7 ((π‘₯ = 𝐽 ∧ 𝑦 = 𝐿) β†’ (βˆͺ π‘₯ FilMap 𝑓) = (βˆͺ 𝐽 FilMap 𝑓))
10 simpr 484 . . . . . . 7 ((π‘₯ = 𝐽 ∧ 𝑦 = 𝐿) β†’ 𝑦 = 𝐿)
119, 10fveq12d 6898 . . . . . 6 ((π‘₯ = 𝐽 ∧ 𝑦 = 𝐿) β†’ ((βˆͺ π‘₯ FilMap 𝑓)β€˜π‘¦) = ((βˆͺ 𝐽 FilMap 𝑓)β€˜πΏ))
127, 11oveq12d 7432 . . . . 5 ((π‘₯ = 𝐽 ∧ 𝑦 = 𝐿) β†’ (π‘₯ fLim ((βˆͺ π‘₯ FilMap 𝑓)β€˜π‘¦)) = (𝐽 fLim ((βˆͺ 𝐽 FilMap 𝑓)β€˜πΏ)))
136, 12mpteq12dv 5233 . . . 4 ((π‘₯ = 𝐽 ∧ 𝑦 = 𝐿) β†’ (𝑓 ∈ (βˆͺ π‘₯ ↑m βˆͺ 𝑦) ↦ (π‘₯ fLim ((βˆͺ π‘₯ FilMap 𝑓)β€˜π‘¦))) = (𝑓 ∈ (βˆͺ 𝐽 ↑m βˆͺ 𝐿) ↦ (𝐽 fLim ((βˆͺ 𝐽 FilMap 𝑓)β€˜πΏ))))
14 df-flf 23831 . . . 4 fLimf = (π‘₯ ∈ Top, 𝑦 ∈ βˆͺ ran Fil ↦ (𝑓 ∈ (βˆͺ π‘₯ ↑m βˆͺ 𝑦) ↦ (π‘₯ fLim ((βˆͺ π‘₯ FilMap 𝑓)β€˜π‘¦))))
15 ovex 7447 . . . . 5 (βˆͺ 𝐽 ↑m βˆͺ 𝐿) ∈ V
1615mptex 7229 . . . 4 (𝑓 ∈ (βˆͺ 𝐽 ↑m βˆͺ 𝐿) ↦ (𝐽 fLim ((βˆͺ 𝐽 FilMap 𝑓)β€˜πΏ))) ∈ V
1713, 14, 16ovmpoa 7570 . . 3 ((𝐽 ∈ Top ∧ 𝐿 ∈ βˆͺ ran Fil) β†’ (𝐽 fLimf 𝐿) = (𝑓 ∈ (βˆͺ 𝐽 ↑m βˆͺ 𝐿) ↦ (𝐽 fLim ((βˆͺ 𝐽 FilMap 𝑓)β€˜πΏ))))
181, 3, 17syl2an 595 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ (𝐽 fLimf 𝐿) = (𝑓 ∈ (βˆͺ 𝐽 ↑m βˆͺ 𝐿) ↦ (𝐽 fLim ((βˆͺ 𝐽 FilMap 𝑓)β€˜πΏ))))
19 toponuni 22803 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
2019eqcomd 2733 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ βˆͺ 𝐽 = 𝑋)
21 filunibas 23772 . . . 4 (𝐿 ∈ (Filβ€˜π‘Œ) β†’ βˆͺ 𝐿 = π‘Œ)
2220, 21oveqan12d 7433 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ (βˆͺ 𝐽 ↑m βˆͺ 𝐿) = (𝑋 ↑m π‘Œ))
2320adantr 480 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ βˆͺ 𝐽 = 𝑋)
2423oveq1d 7429 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ (βˆͺ 𝐽 FilMap 𝑓) = (𝑋 FilMap 𝑓))
2524fveq1d 6893 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ ((βˆͺ 𝐽 FilMap 𝑓)β€˜πΏ) = ((𝑋 FilMap 𝑓)β€˜πΏ))
2625oveq2d 7430 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ (𝐽 fLim ((βˆͺ 𝐽 FilMap 𝑓)β€˜πΏ)) = (𝐽 fLim ((𝑋 FilMap 𝑓)β€˜πΏ)))
2722, 26mpteq12dv 5233 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ (𝑓 ∈ (βˆͺ 𝐽 ↑m βˆͺ 𝐿) ↦ (𝐽 fLim ((βˆͺ 𝐽 FilMap 𝑓)β€˜πΏ))) = (𝑓 ∈ (𝑋 ↑m π‘Œ) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)β€˜πΏ))))
2818, 27eqtrd 2767 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ (𝐽 fLimf 𝐿) = (𝑓 ∈ (𝑋 ↑m π‘Œ) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)β€˜πΏ))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆͺ cuni 4903   ↦ cmpt 5225  ran crn 5673  β€˜cfv 6542  (class class class)co 7414   ↑m cmap 8836  Topctop 22782  TopOnctopon 22799  Filcfil 23736   FilMap cfm 23824   fLim cflim 23825   fLimf cflf 23826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-fbas 21263  df-topon 22800  df-fil 23737  df-flf 23831
This theorem is referenced by:  flfval  23881
  Copyright terms: Public domain W3C validator