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Theorem topnval 16696
Description: Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.)
Hypotheses
Ref Expression
topnval.1 𝐵 = (Base‘𝑊)
topnval.2 𝐽 = (TopSet‘𝑊)
Assertion
Ref Expression
topnval (𝐽t 𝐵) = (TopOpen‘𝑊)

Proof of Theorem topnval
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6663 . . . . . 6 (𝑤 = 𝑊 → (TopSet‘𝑤) = (TopSet‘𝑊))
2 topnval.2 . . . . . 6 𝐽 = (TopSet‘𝑊)
31, 2syl6eqr 2871 . . . . 5 (𝑤 = 𝑊 → (TopSet‘𝑤) = 𝐽)
4 fveq2 6663 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
5 topnval.1 . . . . . 6 𝐵 = (Base‘𝑊)
64, 5syl6eqr 2871 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
73, 6oveq12d 7163 . . . 4 (𝑤 = 𝑊 → ((TopSet‘𝑤) ↾t (Base‘𝑤)) = (𝐽t 𝐵))
8 df-topn 16685 . . . 4 TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤)))
9 ovex 7178 . . . 4 (𝐽t 𝐵) ∈ V
107, 8, 9fvmpt 6761 . . 3 (𝑊 ∈ V → (TopOpen‘𝑊) = (𝐽t 𝐵))
1110eqcomd 2824 . 2 (𝑊 ∈ V → (𝐽t 𝐵) = (TopOpen‘𝑊))
12 0rest 16691 . . 3 (∅ ↾t 𝐵) = ∅
13 fvprc 6656 . . . . 5 𝑊 ∈ V → (TopSet‘𝑊) = ∅)
142, 13syl5eq 2865 . . . 4 𝑊 ∈ V → 𝐽 = ∅)
1514oveq1d 7160 . . 3 𝑊 ∈ V → (𝐽t 𝐵) = (∅ ↾t 𝐵))
16 fvprc 6656 . . 3 𝑊 ∈ V → (TopOpen‘𝑊) = ∅)
1712, 15, 163eqtr4a 2879 . 2 𝑊 ∈ V → (𝐽t 𝐵) = (TopOpen‘𝑊))
1811, 17pm2.61i 183 1 (𝐽t 𝐵) = (TopOpen‘𝑊)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1528  wcel 2105  Vcvv 3492  c0 4288  cfv 6348  (class class class)co 7145  Basecbs 16471  TopSetcts 16559  t crest 16682  TopOpenctopn 16683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-rest 16684  df-topn 16685
This theorem is referenced by:  topnid  16697  topnpropd  16698  oppgtopn  18419  symgtopn  18463  mgptopn  19177  resstopn  21722  prdstopn  22164  tuslem  22803  xrge0tsms  23369  om1opn  23567  xrge0tsmsd  30619  xrge0tmdALT  31088  efmndtopn  43981
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