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Theorem topnval 16700
 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 2872 . . . . 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 2872 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
73, 6oveq12d 7166 . . . 4 (𝑤 = 𝑊 → ((TopSet‘𝑤) ↾t (Base‘𝑤)) = (𝐽t 𝐵))
8 df-topn 16689 . . . 4 TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤)))
9 ovex 7181 . . . 4 (𝐽t 𝐵) ∈ V
107, 8, 9fvmpt 6761 . . 3 (𝑊 ∈ V → (TopOpen‘𝑊) = (𝐽t 𝐵))
1110eqcomd 2825 . 2 (𝑊 ∈ V → (𝐽t 𝐵) = (TopOpen‘𝑊))
12 0rest 16695 . . 3 (∅ ↾t 𝐵) = ∅
13 fvprc 6656 . . . . 5 𝑊 ∈ V → (TopSet‘𝑊) = ∅)
142, 13syl5eq 2866 . . . 4 𝑊 ∈ V → 𝐽 = ∅)
1514oveq1d 7163 . . 3 𝑊 ∈ V → (𝐽t 𝐵) = (∅ ↾t 𝐵))
16 fvprc 6656 . . 3 𝑊 ∈ V → (TopOpen‘𝑊) = ∅)
1712, 15, 163eqtr4a 2880 . 2 𝑊 ∈ V → (𝐽t 𝐵) = (TopOpen‘𝑊))
1811, 17pm2.61i 184 1 (𝐽t 𝐵) = (TopOpen‘𝑊)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1530   ∈ wcel 2107  Vcvv 3493  ∅c0 4289  ‘cfv 6348  (class class class)co 7148  Basecbs 16475  TopSetcts 16563   ↾t crest 16686  TopOpenctopn 16687 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  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 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-rest 16688  df-topn 16689 This theorem is referenced by:  topnid  16701  topnpropd  16702  efmndtopn  18040  oppgtopn  18473  symgtopn  18526  mgptopn  19240  resstopn  21786  prdstopn  22228  tuslem  22868  xrge0tsms  23434  om1opn  23632  xrge0tsmsd  30685  xrge0tmdALT  31177
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