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Theorem topnvalg 12146
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 2697 . . 3 (𝑊𝑉𝑊 ∈ V)
2 restfn 12138 . . . 4 t Fn (V × V)
3 topnval.2 . . . . 5 𝐽 = (TopSet‘𝑊)
4 tsetslid 12123 . . . . . 6 (TopSet = Slot (TopSet‘ndx) ∧ (TopSet‘ndx) ∈ ℕ)
54slotex 12000 . . . . 5 (𝑊𝑉 → (TopSet‘𝑊) ∈ V)
63, 5eqeltrid 2226 . . . 4 (𝑊𝑉𝐽 ∈ V)
7 topnval.1 . . . . 5 𝐵 = (Base‘𝑊)
8 baseslid 12029 . . . . . 6 (Base = Slot (Base‘ndx) ∧ (Base‘ndx) ∈ ℕ)
98slotex 12000 . . . . 5 (𝑊𝑉 → (Base‘𝑊) ∈ V)
107, 9eqeltrid 2226 . . . 4 (𝑊𝑉𝐵 ∈ V)
11 fnovex 5804 . . . 4 (( ↾t Fn (V × V) ∧ 𝐽 ∈ V ∧ 𝐵 ∈ V) → (𝐽t 𝐵) ∈ V)
122, 6, 10, 11mp3an2i 1320 . . 3 (𝑊𝑉 → (𝐽t 𝐵) ∈ V)
13 fveq2 5421 . . . . . 6 (𝑤 = 𝑊 → (TopSet‘𝑤) = (TopSet‘𝑊))
1413, 3eqtr4di 2190 . . . . 5 (𝑤 = 𝑊 → (TopSet‘𝑤) = 𝐽)
15 fveq2 5421 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
1615, 7eqtr4di 2190 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
1714, 16oveq12d 5792 . . . 4 (𝑤 = 𝑊 → ((TopSet‘𝑤) ↾t (Base‘𝑤)) = (𝐽t 𝐵))
18 df-topn 12137 . . . 4 TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤)))
1917, 18fvmptg 5497 . . 3 ((𝑊 ∈ V ∧ (𝐽t 𝐵) ∈ V) → (TopOpen‘𝑊) = (𝐽t 𝐵))
201, 12, 19syl2anc 408 . 2 (𝑊𝑉 → (TopOpen‘𝑊) = (𝐽t 𝐵))
2120eqcomd 2145 1 (𝑊𝑉 → (𝐽t 𝐵) = (TopOpen‘𝑊))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  wcel 1480  Vcvv 2686   × cxp 4537   Fn wfn 5118  cfv 5123  (class class class)co 5774  Basecbs 11973  TopSetcts 12041  t crest 12134  TopOpenctopn 12135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-cnex 7723  ax-resscn 7724  ax-1re 7726  ax-addrcl 7729
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-inn 8733  df-2 8791  df-3 8792  df-4 8793  df-5 8794  df-6 8795  df-7 8796  df-8 8797  df-9 8798  df-ndx 11976  df-slot 11977  df-base 11979  df-tset 12054  df-rest 12136  df-topn 12137
This theorem is referenced by:  topnidg  12147  topnpropgd  12148
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