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Theorem flffval 21706
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 20640 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 fvssunirn 6176 . . . 4 (Fil‘𝑌) ⊆ ran Fil
32sseli 3580 . . 3 (𝐿 ∈ (Fil‘𝑌) → 𝐿 ran Fil)
4 unieq 4412 . . . . . 6 (𝑥 = 𝐽 𝑥 = 𝐽)
5 unieq 4412 . . . . . 6 (𝑦 = 𝐿 𝑦 = 𝐿)
64, 5oveqan12d 6626 . . . . 5 ((𝑥 = 𝐽𝑦 = 𝐿) → ( 𝑥𝑚 𝑦) = ( 𝐽𝑚 𝐿))
7 simpl 473 . . . . . 6 ((𝑥 = 𝐽𝑦 = 𝐿) → 𝑥 = 𝐽)
84adantr 481 . . . . . . . 8 ((𝑥 = 𝐽𝑦 = 𝐿) → 𝑥 = 𝐽)
98oveq1d 6622 . . . . . . 7 ((𝑥 = 𝐽𝑦 = 𝐿) → ( 𝑥 FilMap 𝑓) = ( 𝐽 FilMap 𝑓))
10 simpr 477 . . . . . . 7 ((𝑥 = 𝐽𝑦 = 𝐿) → 𝑦 = 𝐿)
119, 10fveq12d 6156 . . . . . 6 ((𝑥 = 𝐽𝑦 = 𝐿) → (( 𝑥 FilMap 𝑓)‘𝑦) = (( 𝐽 FilMap 𝑓)‘𝐿))
127, 11oveq12d 6625 . . . . 5 ((𝑥 = 𝐽𝑦 = 𝐿) → (𝑥 fLim (( 𝑥 FilMap 𝑓)‘𝑦)) = (𝐽 fLim (( 𝐽 FilMap 𝑓)‘𝐿)))
136, 12mpteq12dv 4695 . . . 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 21657 . . . 4 fLimf = (𝑥 ∈ Top, 𝑦 ran Fil ↦ (𝑓 ∈ ( 𝑥𝑚 𝑦) ↦ (𝑥 fLim (( 𝑥 FilMap 𝑓)‘𝑦))))
15 ovex 6635 . . . . 5 ( 𝐽𝑚 𝐿) ∈ V
1615mptex 6443 . . . 4 (𝑓 ∈ ( 𝐽𝑚 𝐿) ↦ (𝐽 fLim (( 𝐽 FilMap 𝑓)‘𝐿))) ∈ V
1713, 14, 16ovmpt2a 6747 . . 3 ((𝐽 ∈ Top ∧ 𝐿 ran Fil) → (𝐽 fLimf 𝐿) = (𝑓 ∈ ( 𝐽𝑚 𝐿) ↦ (𝐽 fLim (( 𝐽 FilMap 𝑓)‘𝐿))))
181, 3, 17syl2an 494 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLimf 𝐿) = (𝑓 ∈ ( 𝐽𝑚 𝐿) ↦ (𝐽 fLim (( 𝐽 FilMap 𝑓)‘𝐿))))
19 toponuni 20641 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
2019eqcomd 2627 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 = 𝑋)
21 filunibas 21598 . . . 4 (𝐿 ∈ (Fil‘𝑌) → 𝐿 = 𝑌)
2220, 21oveqan12d 6626 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → ( 𝐽𝑚 𝐿) = (𝑋𝑚 𝑌))
2320adantr 481 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → 𝐽 = 𝑋)
2423oveq1d 6622 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → ( 𝐽 FilMap 𝑓) = (𝑋 FilMap 𝑓))
2524fveq1d 6152 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (( 𝐽 FilMap 𝑓)‘𝐿) = ((𝑋 FilMap 𝑓)‘𝐿))
2625oveq2d 6623 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLim (( 𝐽 FilMap 𝑓)‘𝐿)) = (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))
2722, 26mpteq12dv 4695 . 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 2655 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 384   = wceq 1480  wcel 1987   cuni 4404  cmpt 4675  ran crn 5077  cfv 5849  (class class class)co 6607  𝑚 cmap 7805  Topctop 20620  TopOnctopon 20637  Filcfil 21562   FilMap cfm 21650   fLim cflim 21651   fLimf cflf 21652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-fbas 19665  df-topon 20638  df-fil 21563  df-flf 21657
This theorem is referenced by:  flfval  21707
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