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Theorem flffval 23923
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 22845 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2 fvssunirn 6927 . . . 4 (Filβ€˜π‘Œ) βŠ† βˆͺ ran Fil
32sseli 3973 . . 3 (𝐿 ∈ (Filβ€˜π‘Œ) β†’ 𝐿 ∈ βˆͺ ran Fil)
4 unieq 4919 . . . . . 6 (π‘₯ = 𝐽 β†’ βˆͺ π‘₯ = βˆͺ 𝐽)
5 unieq 4919 . . . . . 6 (𝑦 = 𝐿 β†’ βˆͺ 𝑦 = βˆͺ 𝐿)
64, 5oveqan12d 7436 . . . . 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 7432 . . . . . . 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 6901 . . . . . 6 ((π‘₯ = 𝐽 ∧ 𝑦 = 𝐿) β†’ ((βˆͺ π‘₯ FilMap 𝑓)β€˜π‘¦) = ((βˆͺ 𝐽 FilMap 𝑓)β€˜πΏ))
127, 11oveq12d 7435 . . . . 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 5239 . . . 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 23874 . . . 4 fLimf = (π‘₯ ∈ Top, 𝑦 ∈ βˆͺ ran Fil ↦ (𝑓 ∈ (βˆͺ π‘₯ ↑m βˆͺ 𝑦) ↦ (π‘₯ fLim ((βˆͺ π‘₯ FilMap 𝑓)β€˜π‘¦))))
15 ovex 7450 . . . . 5 (βˆͺ 𝐽 ↑m βˆͺ 𝐿) ∈ V
1615mptex 7233 . . . 4 (𝑓 ∈ (βˆͺ 𝐽 ↑m βˆͺ 𝐿) ↦ (𝐽 fLim ((βˆͺ 𝐽 FilMap 𝑓)β€˜πΏ))) ∈ V
1713, 14, 16ovmpoa 7574 . . 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 22846 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
2019eqcomd 2731 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ βˆͺ 𝐽 = 𝑋)
21 filunibas 23815 . . . 4 (𝐿 ∈ (Filβ€˜π‘Œ) β†’ βˆͺ 𝐿 = π‘Œ)
2220, 21oveqan12d 7436 . . 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 7432 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ (βˆͺ 𝐽 FilMap 𝑓) = (𝑋 FilMap 𝑓))
2524fveq1d 6896 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ ((βˆͺ 𝐽 FilMap 𝑓)β€˜πΏ) = ((𝑋 FilMap 𝑓)β€˜πΏ))
2625oveq2d 7433 . . 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 5239 . 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 2765 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 1533   ∈ wcel 2098  βˆͺ cuni 4908   ↦ cmpt 5231  ran crn 5678  β€˜cfv 6547  (class class class)co 7417   ↑m cmap 8843  Topctop 22825  TopOnctopon 22842  Filcfil 23779   FilMap cfm 23867   fLim cflim 23868   fLimf cflf 23869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  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 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-ov 7420  df-oprab 7421  df-mpo 7422  df-fbas 21280  df-topon 22843  df-fil 23780  df-flf 23874
This theorem is referenced by:  flfval  23924
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