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Theorem flffval 24013
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 22935 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 fvssunirn 6940 . . . 4 (Fil‘𝑌) ⊆ ran Fil
32sseli 3991 . . 3 (𝐿 ∈ (Fil‘𝑌) → 𝐿 ran Fil)
4 unieq 4923 . . . . . 6 (𝑥 = 𝐽 𝑥 = 𝐽)
5 unieq 4923 . . . . . 6 (𝑦 = 𝐿 𝑦 = 𝐿)
64, 5oveqan12d 7450 . . . . 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 7446 . . . . . . 7 ((𝑥 = 𝐽𝑦 = 𝐿) → ( 𝑥 FilMap 𝑓) = ( 𝐽 FilMap 𝑓))
10 simpr 484 . . . . . . 7 ((𝑥 = 𝐽𝑦 = 𝐿) → 𝑦 = 𝐿)
119, 10fveq12d 6914 . . . . . 6 ((𝑥 = 𝐽𝑦 = 𝐿) → (( 𝑥 FilMap 𝑓)‘𝑦) = (( 𝐽 FilMap 𝑓)‘𝐿))
127, 11oveq12d 7449 . . . . 5 ((𝑥 = 𝐽𝑦 = 𝐿) → (𝑥 fLim (( 𝑥 FilMap 𝑓)‘𝑦)) = (𝐽 fLim (( 𝐽 FilMap 𝑓)‘𝐿)))
136, 12mpteq12dv 5239 . . . 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 23964 . . . 4 fLimf = (𝑥 ∈ Top, 𝑦 ran Fil ↦ (𝑓 ∈ ( 𝑥m 𝑦) ↦ (𝑥 fLim (( 𝑥 FilMap 𝑓)‘𝑦))))
15 ovex 7464 . . . . 5 ( 𝐽m 𝐿) ∈ V
1615mptex 7243 . . . 4 (𝑓 ∈ ( 𝐽m 𝐿) ↦ (𝐽 fLim (( 𝐽 FilMap 𝑓)‘𝐿))) ∈ V
1713, 14, 16ovmpoa 7588 . . 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 22936 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
2019eqcomd 2741 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 = 𝑋)
21 filunibas 23905 . . . 4 (𝐿 ∈ (Fil‘𝑌) → 𝐿 = 𝑌)
2220, 21oveqan12d 7450 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → ( 𝐽m 𝐿) = (𝑋m 𝑌))
2320adantr 480 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → 𝐽 = 𝑋)
2423oveq1d 7446 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → ( 𝐽 FilMap 𝑓) = (𝑋 FilMap 𝑓))
2524fveq1d 6909 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (( 𝐽 FilMap 𝑓)‘𝐿) = ((𝑋 FilMap 𝑓)‘𝐿))
2625oveq2d 7447 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLim (( 𝐽 FilMap 𝑓)‘𝐿)) = (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))
2722, 26mpteq12dv 5239 . 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 2775 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 1537  wcel 2106   cuni 4912  cmpt 5231  ran crn 5690  cfv 6563  (class class class)co 7431  m cmap 8865  Topctop 22915  TopOnctopon 22932  Filcfil 23869   FilMap cfm 23957   fLim cflim 23958   fLimf cflf 23959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21379  df-topon 22933  df-fil 23870  df-flf 23964
This theorem is referenced by:  flfval  24014
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