Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  flffval Structured version   Visualization version   GIF version

Theorem flffval 22694
 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 21618 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 fvssunirn 6691 . . . 4 (Fil‘𝑌) ⊆ ran Fil
32sseli 3890 . . 3 (𝐿 ∈ (Fil‘𝑌) → 𝐿 ran Fil)
4 unieq 4812 . . . . . 6 (𝑥 = 𝐽 𝑥 = 𝐽)
5 unieq 4812 . . . . . 6 (𝑦 = 𝐿 𝑦 = 𝐿)
64, 5oveqan12d 7174 . . . . 5 ((𝑥 = 𝐽𝑦 = 𝐿) → ( 𝑥m 𝑦) = ( 𝐽m 𝐿))
7 simpl 486 . . . . . 6 ((𝑥 = 𝐽𝑦 = 𝐿) → 𝑥 = 𝐽)
84adantr 484 . . . . . . . 8 ((𝑥 = 𝐽𝑦 = 𝐿) → 𝑥 = 𝐽)
98oveq1d 7170 . . . . . . 7 ((𝑥 = 𝐽𝑦 = 𝐿) → ( 𝑥 FilMap 𝑓) = ( 𝐽 FilMap 𝑓))
10 simpr 488 . . . . . . 7 ((𝑥 = 𝐽𝑦 = 𝐿) → 𝑦 = 𝐿)
119, 10fveq12d 6669 . . . . . 6 ((𝑥 = 𝐽𝑦 = 𝐿) → (( 𝑥 FilMap 𝑓)‘𝑦) = (( 𝐽 FilMap 𝑓)‘𝐿))
127, 11oveq12d 7173 . . . . 5 ((𝑥 = 𝐽𝑦 = 𝐿) → (𝑥 fLim (( 𝑥 FilMap 𝑓)‘𝑦)) = (𝐽 fLim (( 𝐽 FilMap 𝑓)‘𝐿)))
136, 12mpteq12dv 5120 . . . 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 22645 . . . 4 fLimf = (𝑥 ∈ Top, 𝑦 ran Fil ↦ (𝑓 ∈ ( 𝑥m 𝑦) ↦ (𝑥 fLim (( 𝑥 FilMap 𝑓)‘𝑦))))
15 ovex 7188 . . . . 5 ( 𝐽m 𝐿) ∈ V
1615mptex 6982 . . . 4 (𝑓 ∈ ( 𝐽m 𝐿) ↦ (𝐽 fLim (( 𝐽 FilMap 𝑓)‘𝐿))) ∈ V
1713, 14, 16ovmpoa 7305 . . 3 ((𝐽 ∈ Top ∧ 𝐿 ran Fil) → (𝐽 fLimf 𝐿) = (𝑓 ∈ ( 𝐽m 𝐿) ↦ (𝐽 fLim (( 𝐽 FilMap 𝑓)‘𝐿))))
181, 3, 17syl2an 598 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLimf 𝐿) = (𝑓 ∈ ( 𝐽m 𝐿) ↦ (𝐽 fLim (( 𝐽 FilMap 𝑓)‘𝐿))))
19 toponuni 21619 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
2019eqcomd 2764 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 = 𝑋)
21 filunibas 22586 . . . 4 (𝐿 ∈ (Fil‘𝑌) → 𝐿 = 𝑌)
2220, 21oveqan12d 7174 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → ( 𝐽m 𝐿) = (𝑋m 𝑌))
2320adantr 484 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → 𝐽 = 𝑋)
2423oveq1d 7170 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → ( 𝐽 FilMap 𝑓) = (𝑋 FilMap 𝑓))
2524fveq1d 6664 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (( 𝐽 FilMap 𝑓)‘𝐿) = ((𝑋 FilMap 𝑓)‘𝐿))
2625oveq2d 7171 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLim (( 𝐽 FilMap 𝑓)‘𝐿)) = (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))
2722, 26mpteq12dv 5120 . 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 2793 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 399   = wceq 1538   ∈ wcel 2111  ∪ cuni 4801   ↦ cmpt 5115  ran crn 5528  ‘cfv 6339  (class class class)co 7155   ↑m cmap 8421  Topctop 21598  TopOnctopon 21615  Filcfil 22550   FilMap cfm 22638   fLim cflim 22639   fLimf cflf 22640 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7158  df-oprab 7159  df-mpo 7160  df-fbas 20168  df-topon 21616  df-fil 22551  df-flf 22645 This theorem is referenced by:  flfval  22695
 Copyright terms: Public domain W3C validator