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Theorem topnvalg 12756
Description: Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by Jim Kingdon, 11-Feb-2023.)
Hypotheses
Ref Expression
topnval.1 𝐵 = (Base‘𝑊)
topnval.2 𝐽 = (TopSet‘𝑊)
Assertion
Ref Expression
topnvalg (𝑊𝑉 → (𝐽t 𝐵) = (TopOpen‘𝑊))

Proof of Theorem topnvalg
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elex 2763 . . 3 (𝑊𝑉𝑊 ∈ V)
2 restfn 12748 . . . 4 t Fn (V × V)
3 topnval.2 . . . . 5 𝐽 = (TopSet‘𝑊)
4 tsetslid 12699 . . . . . 6 (TopSet = Slot (TopSet‘ndx) ∧ (TopSet‘ndx) ∈ ℕ)
54slotex 12539 . . . . 5 (𝑊𝑉 → (TopSet‘𝑊) ∈ V)
63, 5eqeltrid 2276 . . . 4 (𝑊𝑉𝐽 ∈ V)
7 topnval.1 . . . . 5 𝐵 = (Base‘𝑊)
8 baseslid 12569 . . . . . 6 (Base = Slot (Base‘ndx) ∧ (Base‘ndx) ∈ ℕ)
98slotex 12539 . . . . 5 (𝑊𝑉 → (Base‘𝑊) ∈ V)
107, 9eqeltrid 2276 . . . 4 (𝑊𝑉𝐵 ∈ V)
11 fnovex 5929 . . . 4 (( ↾t Fn (V × V) ∧ 𝐽 ∈ V ∧ 𝐵 ∈ V) → (𝐽t 𝐵) ∈ V)
122, 6, 10, 11mp3an2i 1353 . . 3 (𝑊𝑉 → (𝐽t 𝐵) ∈ V)
13 fveq2 5534 . . . . . 6 (𝑤 = 𝑊 → (TopSet‘𝑤) = (TopSet‘𝑊))
1413, 3eqtr4di 2240 . . . . 5 (𝑤 = 𝑊 → (TopSet‘𝑤) = 𝐽)
15 fveq2 5534 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
1615, 7eqtr4di 2240 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
1714, 16oveq12d 5914 . . . 4 (𝑤 = 𝑊 → ((TopSet‘𝑤) ↾t (Base‘𝑤)) = (𝐽t 𝐵))
18 df-topn 12747 . . . 4 TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤)))
1917, 18fvmptg 5613 . . 3 ((𝑊 ∈ V ∧ (𝐽t 𝐵) ∈ V) → (TopOpen‘𝑊) = (𝐽t 𝐵))
201, 12, 19syl2anc 411 . 2 (𝑊𝑉 → (TopOpen‘𝑊) = (𝐽t 𝐵))
2120eqcomd 2195 1 (𝑊𝑉 → (𝐽t 𝐵) = (TopOpen‘𝑊))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wcel 2160  Vcvv 2752   × cxp 4642   Fn wfn 5230  cfv 5235  (class class class)co 5896  Basecbs 12512  TopSetcts 12595  t crest 12744  TopOpenctopn 12745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-cnex 7932  ax-resscn 7933  ax-1re 7935  ax-addrcl 7938
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-inn 8950  df-2 9008  df-3 9009  df-4 9010  df-5 9011  df-6 9012  df-7 9013  df-8 9014  df-9 9015  df-ndx 12515  df-slot 12516  df-base 12518  df-tset 12608  df-rest 12746  df-topn 12747
This theorem is referenced by:  topnidg  12757  topnpropgd  12758  mgptopng  13283
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