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Theorem flffval 23713
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 22635 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2 fvssunirn 6923 . . . 4 (Filβ€˜π‘Œ) βŠ† βˆͺ ran Fil
32sseli 3977 . . 3 (𝐿 ∈ (Filβ€˜π‘Œ) β†’ 𝐿 ∈ βˆͺ ran Fil)
4 unieq 4918 . . . . . 6 (π‘₯ = 𝐽 β†’ βˆͺ π‘₯ = βˆͺ 𝐽)
5 unieq 4918 . . . . . 6 (𝑦 = 𝐿 β†’ βˆͺ 𝑦 = βˆͺ 𝐿)
64, 5oveqan12d 7430 . . . . 5 ((π‘₯ = 𝐽 ∧ 𝑦 = 𝐿) β†’ (βˆͺ π‘₯ ↑m βˆͺ 𝑦) = (βˆͺ 𝐽 ↑m βˆͺ 𝐿))
7 simpl 481 . . . . . 6 ((π‘₯ = 𝐽 ∧ 𝑦 = 𝐿) β†’ π‘₯ = 𝐽)
84adantr 479 . . . . . . . 8 ((π‘₯ = 𝐽 ∧ 𝑦 = 𝐿) β†’ βˆͺ π‘₯ = βˆͺ 𝐽)
98oveq1d 7426 . . . . . . 7 ((π‘₯ = 𝐽 ∧ 𝑦 = 𝐿) β†’ (βˆͺ π‘₯ FilMap 𝑓) = (βˆͺ 𝐽 FilMap 𝑓))
10 simpr 483 . . . . . . 7 ((π‘₯ = 𝐽 ∧ 𝑦 = 𝐿) β†’ 𝑦 = 𝐿)
119, 10fveq12d 6897 . . . . . 6 ((π‘₯ = 𝐽 ∧ 𝑦 = 𝐿) β†’ ((βˆͺ π‘₯ FilMap 𝑓)β€˜π‘¦) = ((βˆͺ 𝐽 FilMap 𝑓)β€˜πΏ))
127, 11oveq12d 7429 . . . . 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 5238 . . . 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 23664 . . . 4 fLimf = (π‘₯ ∈ Top, 𝑦 ∈ βˆͺ ran Fil ↦ (𝑓 ∈ (βˆͺ π‘₯ ↑m βˆͺ 𝑦) ↦ (π‘₯ fLim ((βˆͺ π‘₯ FilMap 𝑓)β€˜π‘¦))))
15 ovex 7444 . . . . 5 (βˆͺ 𝐽 ↑m βˆͺ 𝐿) ∈ V
1615mptex 7226 . . . 4 (𝑓 ∈ (βˆͺ 𝐽 ↑m βˆͺ 𝐿) ↦ (𝐽 fLim ((βˆͺ 𝐽 FilMap 𝑓)β€˜πΏ))) ∈ V
1713, 14, 16ovmpoa 7565 . . 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 594 . 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 22636 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
2019eqcomd 2736 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ βˆͺ 𝐽 = 𝑋)
21 filunibas 23605 . . . 4 (𝐿 ∈ (Filβ€˜π‘Œ) β†’ βˆͺ 𝐿 = π‘Œ)
2220, 21oveqan12d 7430 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ (βˆͺ 𝐽 ↑m βˆͺ 𝐿) = (𝑋 ↑m π‘Œ))
2320adantr 479 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ βˆͺ 𝐽 = 𝑋)
2423oveq1d 7426 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ (βˆͺ 𝐽 FilMap 𝑓) = (𝑋 FilMap 𝑓))
2524fveq1d 6892 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ ((βˆͺ 𝐽 FilMap 𝑓)β€˜πΏ) = ((𝑋 FilMap 𝑓)β€˜πΏ))
2625oveq2d 7427 . . 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 5238 . 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 2770 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 394   = wceq 1539   ∈ wcel 2104  βˆͺ cuni 4907   ↦ cmpt 5230  ran crn 5676  β€˜cfv 6542  (class class class)co 7411   ↑m cmap 8822  Topctop 22615  TopOnctopon 22632  Filcfil 23569   FilMap cfm 23657   fLim cflim 23658   fLimf cflf 23659
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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
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 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 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-fbas 21141  df-topon 22633  df-fil 23570  df-flf 23664
This theorem is referenced by:  flfval  23714
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