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Theorem topnvalg 12059
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 2671 . . 3 (𝑊𝑉𝑊 ∈ V)
2 restfn 12051 . . . 4 t Fn (V × V)
3 topnval.2 . . . . 5 𝐽 = (TopSet‘𝑊)
4 tsetslid 12036 . . . . . 6 (TopSet = Slot (TopSet‘ndx) ∧ (TopSet‘ndx) ∈ ℕ)
54slotex 11913 . . . . 5 (𝑊𝑉 → (TopSet‘𝑊) ∈ V)
63, 5eqeltrid 2204 . . . 4 (𝑊𝑉𝐽 ∈ V)
7 topnval.1 . . . . 5 𝐵 = (Base‘𝑊)
8 baseslid 11942 . . . . . 6 (Base = Slot (Base‘ndx) ∧ (Base‘ndx) ∈ ℕ)
98slotex 11913 . . . . 5 (𝑊𝑉 → (Base‘𝑊) ∈ V)
107, 9eqeltrid 2204 . . . 4 (𝑊𝑉𝐵 ∈ V)
11 fnovex 5772 . . . 4 (( ↾t Fn (V × V) ∧ 𝐽 ∈ V ∧ 𝐵 ∈ V) → (𝐽t 𝐵) ∈ V)
122, 6, 10, 11mp3an2i 1305 . . 3 (𝑊𝑉 → (𝐽t 𝐵) ∈ V)
13 fveq2 5389 . . . . . 6 (𝑤 = 𝑊 → (TopSet‘𝑤) = (TopSet‘𝑊))
1413, 3syl6eqr 2168 . . . . 5 (𝑤 = 𝑊 → (TopSet‘𝑤) = 𝐽)
15 fveq2 5389 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
1615, 7syl6eqr 2168 . . . . 5 (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
1714, 16oveq12d 5760 . . . 4 (𝑤 = 𝑊 → ((TopSet‘𝑤) ↾t (Base‘𝑤)) = (𝐽t 𝐵))
18 df-topn 12050 . . . 4 TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤)))
1917, 18fvmptg 5465 . . 3 ((𝑊 ∈ V ∧ (𝐽t 𝐵) ∈ V) → (TopOpen‘𝑊) = (𝐽t 𝐵))
201, 12, 19syl2anc 408 . 2 (𝑊𝑉 → (TopOpen‘𝑊) = (𝐽t 𝐵))
2120eqcomd 2123 1 (𝑊𝑉 → (𝐽t 𝐵) = (TopOpen‘𝑊))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  wcel 1465  Vcvv 2660   × cxp 4507   Fn wfn 5088  cfv 5093  (class class class)co 5742  Basecbs 11886  TopSetcts 11954  t crest 12047  TopOpenctopn 12048
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-cnex 7679  ax-resscn 7680  ax-1re 7682  ax-addrcl 7685
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-inn 8689  df-2 8747  df-3 8748  df-4 8749  df-5 8750  df-6 8751  df-7 8752  df-8 8753  df-9 8754  df-ndx 11889  df-slot 11890  df-base 11892  df-tset 11967  df-rest 12049  df-topn 12050
This theorem is referenced by:  topnidg  12060  topnpropgd  12061
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