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Mirrors > Home > ILE Home > Th. List > topnvalg | GIF version |
Description: Value of the topology extractor function. (Contributed by Mario Carneiro, 13-Aug-2015.) (Revised by Jim Kingdon, 11-Feb-2023.) |
Ref | Expression |
---|---|
topnval.1 | ⊢ 𝐵 = (Base‘𝑊) |
topnval.2 | ⊢ 𝐽 = (TopSet‘𝑊) |
Ref | Expression |
---|---|
topnvalg | ⊢ (𝑊 ∈ 𝑉 → (𝐽 ↾t 𝐵) = (TopOpen‘𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2671 | . . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ 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) | |
5 | 4 | slotex 11913 | . . . . 5 ⊢ (𝑊 ∈ 𝑉 → (TopSet‘𝑊) ∈ V) |
6 | 3, 5 | eqeltrid 2204 | . . . 4 ⊢ (𝑊 ∈ 𝑉 → 𝐽 ∈ V) |
7 | topnval.1 | . . . . 5 ⊢ 𝐵 = (Base‘𝑊) | |
8 | baseslid 11942 | . . . . . 6 ⊢ (Base = Slot (Base‘ndx) ∧ (Base‘ndx) ∈ ℕ) | |
9 | 8 | slotex 11913 | . . . . 5 ⊢ (𝑊 ∈ 𝑉 → (Base‘𝑊) ∈ V) |
10 | 7, 9 | eqeltrid 2204 | . . . 4 ⊢ (𝑊 ∈ 𝑉 → 𝐵 ∈ V) |
11 | fnovex 5772 | . . . 4 ⊢ (( ↾t Fn (V × V) ∧ 𝐽 ∈ V ∧ 𝐵 ∈ V) → (𝐽 ↾t 𝐵) ∈ V) | |
12 | 2, 6, 10, 11 | mp3an2i 1305 | . . 3 ⊢ (𝑊 ∈ 𝑉 → (𝐽 ↾t 𝐵) ∈ V) |
13 | fveq2 5389 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = (TopSet‘𝑊)) | |
14 | 13, 3 | syl6eqr 2168 | . . . . 5 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = 𝐽) |
15 | fveq2 5389 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) | |
16 | 15, 7 | syl6eqr 2168 | . . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵) |
17 | 14, 16 | oveq12d 5760 | . . . 4 ⊢ (𝑤 = 𝑊 → ((TopSet‘𝑤) ↾t (Base‘𝑤)) = (𝐽 ↾t 𝐵)) |
18 | df-topn 12050 | . . . 4 ⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤))) | |
19 | 17, 18 | fvmptg 5465 | . . 3 ⊢ ((𝑊 ∈ V ∧ (𝐽 ↾t 𝐵) ∈ V) → (TopOpen‘𝑊) = (𝐽 ↾t 𝐵)) |
20 | 1, 12, 19 | syl2anc 408 | . 2 ⊢ (𝑊 ∈ 𝑉 → (TopOpen‘𝑊) = (𝐽 ↾t 𝐵)) |
21 | 20 | eqcomd 2123 | 1 ⊢ (𝑊 ∈ 𝑉 → (𝐽 ↾t 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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