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Theorem flffval 23998
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 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLimf 𝐿) = (𝑓 ∈ (𝑋m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿))))
Distinct variable groups:   𝑓,𝐽   𝑓,𝑋   𝑓,𝑌   𝑓,𝐿

Proof of Theorem flffval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 topontop 22920 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 fvssunirn 6938 . . . 4 (Fil‘𝑌) ⊆ ran Fil
32sseli 3978 . . 3 (𝐿 ∈ (Fil‘𝑌) → 𝐿 ran Fil)
4 unieq 4917 . . . . . 6 (𝑥 = 𝐽 𝑥 = 𝐽)
5 unieq 4917 . . . . . 6 (𝑦 = 𝐿 𝑦 = 𝐿)
64, 5oveqan12d 7451 . . . . 5 ((𝑥 = 𝐽𝑦 = 𝐿) → ( 𝑥m 𝑦) = ( 𝐽m 𝐿))
7 simpl 482 . . . . . 6 ((𝑥 = 𝐽𝑦 = 𝐿) → 𝑥 = 𝐽)
84adantr 480 . . . . . . . 8 ((𝑥 = 𝐽𝑦 = 𝐿) → 𝑥 = 𝐽)
98oveq1d 7447 . . . . . . 7 ((𝑥 = 𝐽𝑦 = 𝐿) → ( 𝑥 FilMap 𝑓) = ( 𝐽 FilMap 𝑓))
10 simpr 484 . . . . . . 7 ((𝑥 = 𝐽𝑦 = 𝐿) → 𝑦 = 𝐿)
119, 10fveq12d 6912 . . . . . 6 ((𝑥 = 𝐽𝑦 = 𝐿) → (( 𝑥 FilMap 𝑓)‘𝑦) = (( 𝐽 FilMap 𝑓)‘𝐿))
127, 11oveq12d 7450 . . . . 5 ((𝑥 = 𝐽𝑦 = 𝐿) → (𝑥 fLim (( 𝑥 FilMap 𝑓)‘𝑦)) = (𝐽 fLim (( 𝐽 FilMap 𝑓)‘𝐿)))
136, 12mpteq12dv 5232 . . . 4 ((𝑥 = 𝐽𝑦 = 𝐿) → (𝑓 ∈ ( 𝑥m 𝑦) ↦ (𝑥 fLim (( 𝑥 FilMap 𝑓)‘𝑦))) = (𝑓 ∈ ( 𝐽m 𝐿) ↦ (𝐽 fLim (( 𝐽 FilMap 𝑓)‘𝐿))))
14 df-flf 23949 . . . 4 fLimf = (𝑥 ∈ Top, 𝑦 ran Fil ↦ (𝑓 ∈ ( 𝑥m 𝑦) ↦ (𝑥 fLim (( 𝑥 FilMap 𝑓)‘𝑦))))
15 ovex 7465 . . . . 5 ( 𝐽m 𝐿) ∈ V
1615mptex 7244 . . . 4 (𝑓 ∈ ( 𝐽m 𝐿) ↦ (𝐽 fLim (( 𝐽 FilMap 𝑓)‘𝐿))) ∈ V
1713, 14, 16ovmpoa 7589 . . 3 ((𝐽 ∈ Top ∧ 𝐿 ran Fil) → (𝐽 fLimf 𝐿) = (𝑓 ∈ ( 𝐽m 𝐿) ↦ (𝐽 fLim (( 𝐽 FilMap 𝑓)‘𝐿))))
181, 3, 17syl2an 596 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLimf 𝐿) = (𝑓 ∈ ( 𝐽m 𝐿) ↦ (𝐽 fLim (( 𝐽 FilMap 𝑓)‘𝐿))))
19 toponuni 22921 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
2019eqcomd 2742 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 = 𝑋)
21 filunibas 23890 . . . 4 (𝐿 ∈ (Fil‘𝑌) → 𝐿 = 𝑌)
2220, 21oveqan12d 7451 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → ( 𝐽m 𝐿) = (𝑋m 𝑌))
2320adantr 480 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → 𝐽 = 𝑋)
2423oveq1d 7447 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → ( 𝐽 FilMap 𝑓) = (𝑋 FilMap 𝑓))
2524fveq1d 6907 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (( 𝐽 FilMap 𝑓)‘𝐿) = ((𝑋 FilMap 𝑓)‘𝐿))
2625oveq2d 7448 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLim (( 𝐽 FilMap 𝑓)‘𝐿)) = (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))
2722, 26mpteq12dv 5232 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝑓 ∈ ( 𝐽m 𝐿) ↦ (𝐽 fLim (( 𝐽 FilMap 𝑓)‘𝐿))) = (𝑓 ∈ (𝑋m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿))))
2818, 27eqtrd 2776 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLimf 𝐿) = (𝑓 ∈ (𝑋m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107   cuni 4906  cmpt 5224  ran crn 5685  cfv 6560  (class class class)co 7432  m cmap 8867  Topctop 22900  TopOnctopon 22917  Filcfil 23854   FilMap cfm 23942   fLim cflim 23943   fLimf cflf 23944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-fbas 21362  df-topon 22918  df-fil 23855  df-flf 23949
This theorem is referenced by:  flfval  23999
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