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Theorem flffval 23493
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 22415 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2 fvssunirn 6925 . . . 4 (Filβ€˜π‘Œ) βŠ† βˆͺ ran Fil
32sseli 3979 . . 3 (𝐿 ∈ (Filβ€˜π‘Œ) β†’ 𝐿 ∈ βˆͺ ran Fil)
4 unieq 4920 . . . . . 6 (π‘₯ = 𝐽 β†’ βˆͺ π‘₯ = βˆͺ 𝐽)
5 unieq 4920 . . . . . 6 (𝑦 = 𝐿 β†’ βˆͺ 𝑦 = βˆͺ 𝐿)
64, 5oveqan12d 7428 . . . . 5 ((π‘₯ = 𝐽 ∧ 𝑦 = 𝐿) β†’ (βˆͺ π‘₯ ↑m βˆͺ 𝑦) = (βˆͺ 𝐽 ↑m βˆͺ 𝐿))
7 simpl 484 . . . . . 6 ((π‘₯ = 𝐽 ∧ 𝑦 = 𝐿) β†’ π‘₯ = 𝐽)
84adantr 482 . . . . . . . 8 ((π‘₯ = 𝐽 ∧ 𝑦 = 𝐿) β†’ βˆͺ π‘₯ = βˆͺ 𝐽)
98oveq1d 7424 . . . . . . 7 ((π‘₯ = 𝐽 ∧ 𝑦 = 𝐿) β†’ (βˆͺ π‘₯ FilMap 𝑓) = (βˆͺ 𝐽 FilMap 𝑓))
10 simpr 486 . . . . . . 7 ((π‘₯ = 𝐽 ∧ 𝑦 = 𝐿) β†’ 𝑦 = 𝐿)
119, 10fveq12d 6899 . . . . . 6 ((π‘₯ = 𝐽 ∧ 𝑦 = 𝐿) β†’ ((βˆͺ π‘₯ FilMap 𝑓)β€˜π‘¦) = ((βˆͺ 𝐽 FilMap 𝑓)β€˜πΏ))
127, 11oveq12d 7427 . . . . 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 5240 . . . 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 23444 . . . 4 fLimf = (π‘₯ ∈ Top, 𝑦 ∈ βˆͺ ran Fil ↦ (𝑓 ∈ (βˆͺ π‘₯ ↑m βˆͺ 𝑦) ↦ (π‘₯ fLim ((βˆͺ π‘₯ FilMap 𝑓)β€˜π‘¦))))
15 ovex 7442 . . . . 5 (βˆͺ 𝐽 ↑m βˆͺ 𝐿) ∈ V
1615mptex 7225 . . . 4 (𝑓 ∈ (βˆͺ 𝐽 ↑m βˆͺ 𝐿) ↦ (𝐽 fLim ((βˆͺ 𝐽 FilMap 𝑓)β€˜πΏ))) ∈ V
1713, 14, 16ovmpoa 7563 . . 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 597 . 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 22416 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
2019eqcomd 2739 . . . 4 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ βˆͺ 𝐽 = 𝑋)
21 filunibas 23385 . . . 4 (𝐿 ∈ (Filβ€˜π‘Œ) β†’ βˆͺ 𝐿 = π‘Œ)
2220, 21oveqan12d 7428 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ (βˆͺ 𝐽 ↑m βˆͺ 𝐿) = (𝑋 ↑m π‘Œ))
2320adantr 482 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ βˆͺ 𝐽 = 𝑋)
2423oveq1d 7424 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ (βˆͺ 𝐽 FilMap 𝑓) = (𝑋 FilMap 𝑓))
2524fveq1d 6894 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐿 ∈ (Filβ€˜π‘Œ)) β†’ ((βˆͺ 𝐽 FilMap 𝑓)β€˜πΏ) = ((𝑋 FilMap 𝑓)β€˜πΏ))
2625oveq2d 7425 . . 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 5240 . 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 2773 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 397   = wceq 1542   ∈ wcel 2107  βˆͺ cuni 4909   ↦ cmpt 5232  ran crn 5678  β€˜cfv 6544  (class class class)co 7409   ↑m cmap 8820  Topctop 22395  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-fbas 20941  df-topon 22413  df-fil 23350  df-flf 23444
This theorem is referenced by:  flfval  23494
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