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Theorem flffval 22013
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 20938 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 fvssunirn 6358 . . . 4 (Fil‘𝑌) ⊆ ran Fil
32sseli 3748 . . 3 (𝐿 ∈ (Fil‘𝑌) → 𝐿 ran Fil)
4 unieq 4582 . . . . . 6 (𝑥 = 𝐽 𝑥 = 𝐽)
5 unieq 4582 . . . . . 6 (𝑦 = 𝐿 𝑦 = 𝐿)
64, 5oveqan12d 6812 . . . . 5 ((𝑥 = 𝐽𝑦 = 𝐿) → ( 𝑥𝑚 𝑦) = ( 𝐽𝑚 𝐿))
7 simpl 468 . . . . . 6 ((𝑥 = 𝐽𝑦 = 𝐿) → 𝑥 = 𝐽)
84adantr 466 . . . . . . . 8 ((𝑥 = 𝐽𝑦 = 𝐿) → 𝑥 = 𝐽)
98oveq1d 6808 . . . . . . 7 ((𝑥 = 𝐽𝑦 = 𝐿) → ( 𝑥 FilMap 𝑓) = ( 𝐽 FilMap 𝑓))
10 simpr 471 . . . . . . 7 ((𝑥 = 𝐽𝑦 = 𝐿) → 𝑦 = 𝐿)
119, 10fveq12d 6338 . . . . . 6 ((𝑥 = 𝐽𝑦 = 𝐿) → (( 𝑥 FilMap 𝑓)‘𝑦) = (( 𝐽 FilMap 𝑓)‘𝐿))
127, 11oveq12d 6811 . . . . 5 ((𝑥 = 𝐽𝑦 = 𝐿) → (𝑥 fLim (( 𝑥 FilMap 𝑓)‘𝑦)) = (𝐽 fLim (( 𝐽 FilMap 𝑓)‘𝐿)))
136, 12mpteq12dv 4867 . . . 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 21964 . . . 4 fLimf = (𝑥 ∈ Top, 𝑦 ran Fil ↦ (𝑓 ∈ ( 𝑥𝑚 𝑦) ↦ (𝑥 fLim (( 𝑥 FilMap 𝑓)‘𝑦))))
15 ovex 6823 . . . . 5 ( 𝐽𝑚 𝐿) ∈ V
1615mptex 6630 . . . 4 (𝑓 ∈ ( 𝐽𝑚 𝐿) ↦ (𝐽 fLim (( 𝐽 FilMap 𝑓)‘𝐿))) ∈ V
1713, 14, 16ovmpt2a 6938 . . 3 ((𝐽 ∈ Top ∧ 𝐿 ran Fil) → (𝐽 fLimf 𝐿) = (𝑓 ∈ ( 𝐽𝑚 𝐿) ↦ (𝐽 fLim (( 𝐽 FilMap 𝑓)‘𝐿))))
181, 3, 17syl2an 583 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLimf 𝐿) = (𝑓 ∈ ( 𝐽𝑚 𝐿) ↦ (𝐽 fLim (( 𝐽 FilMap 𝑓)‘𝐿))))
19 toponuni 20939 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
2019eqcomd 2777 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 = 𝑋)
21 filunibas 21905 . . . 4 (𝐿 ∈ (Fil‘𝑌) → 𝐿 = 𝑌)
2220, 21oveqan12d 6812 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → ( 𝐽𝑚 𝐿) = (𝑋𝑚 𝑌))
2320adantr 466 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → 𝐽 = 𝑋)
2423oveq1d 6808 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → ( 𝐽 FilMap 𝑓) = (𝑋 FilMap 𝑓))
2524fveq1d 6334 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (( 𝐽 FilMap 𝑓)‘𝐿) = ((𝑋 FilMap 𝑓)‘𝐿))
2625oveq2d 6809 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLim (( 𝐽 FilMap 𝑓)‘𝐿)) = (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))
2722, 26mpteq12dv 4867 . 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 2805 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 382   = wceq 1631  wcel 2145   cuni 4574  cmpt 4863  ran crn 5250  cfv 6031  (class class class)co 6793  𝑚 cmap 8009  Topctop 20918  TopOnctopon 20935  Filcfil 21869   FilMap cfm 21957   fLim cflim 21958   fLimf cflf 21959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-fbas 19958  df-topon 20936  df-fil 21870  df-flf 21964
This theorem is referenced by:  flfval  22014
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