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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 2697 | . . 3 ⊢ (𝑊 ∈ 𝑉 → 𝑊 ∈ V) | |
2 | restfn 12129 | . . . 4 ⊢ ↾t Fn (V × V) | |
3 | topnval.2 | . . . . 5 ⊢ 𝐽 = (TopSet‘𝑊) | |
4 | tsetslid 12114 | . . . . . 6 ⊢ (TopSet = Slot (TopSet‘ndx) ∧ (TopSet‘ndx) ∈ ℕ) | |
5 | 4 | slotex 11991 | . . . . 5 ⊢ (𝑊 ∈ 𝑉 → (TopSet‘𝑊) ∈ V) |
6 | 3, 5 | eqeltrid 2226 | . . . 4 ⊢ (𝑊 ∈ 𝑉 → 𝐽 ∈ V) |
7 | topnval.1 | . . . . 5 ⊢ 𝐵 = (Base‘𝑊) | |
8 | baseslid 12020 | . . . . . 6 ⊢ (Base = Slot (Base‘ndx) ∧ (Base‘ndx) ∈ ℕ) | |
9 | 8 | slotex 11991 | . . . . 5 ⊢ (𝑊 ∈ 𝑉 → (Base‘𝑊) ∈ V) |
10 | 7, 9 | eqeltrid 2226 | . . . 4 ⊢ (𝑊 ∈ 𝑉 → 𝐵 ∈ V) |
11 | fnovex 5804 | . . . 4 ⊢ (( ↾t Fn (V × V) ∧ 𝐽 ∈ V ∧ 𝐵 ∈ V) → (𝐽 ↾t 𝐵) ∈ V) | |
12 | 2, 6, 10, 11 | mp3an2i 1320 | . . 3 ⊢ (𝑊 ∈ 𝑉 → (𝐽 ↾t 𝐵) ∈ V) |
13 | fveq2 5421 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = (TopSet‘𝑊)) | |
14 | 13, 3 | syl6eqr 2190 | . . . . 5 ⊢ (𝑤 = 𝑊 → (TopSet‘𝑤) = 𝐽) |
15 | fveq2 5421 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊)) | |
16 | 15, 7 | syl6eqr 2190 | . . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵) |
17 | 14, 16 | oveq12d 5792 | . . . 4 ⊢ (𝑤 = 𝑊 → ((TopSet‘𝑤) ↾t (Base‘𝑤)) = (𝐽 ↾t 𝐵)) |
18 | df-topn 12128 | . . . 4 ⊢ TopOpen = (𝑤 ∈ V ↦ ((TopSet‘𝑤) ↾t (Base‘𝑤))) | |
19 | 17, 18 | fvmptg 5497 | . . 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 2145 | 1 ⊢ (𝑊 ∈ 𝑉 → (𝐽 ↾t 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 11964 TopSetcts 12032 ↾t crest 12125 TopOpenctopn 12126 |
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 7716 ax-resscn 7717 ax-1re 7719 ax-addrcl 7722 |
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 8726 df-2 8784 df-3 8785 df-4 8786 df-5 8787 df-6 8788 df-7 8789 df-8 8790 df-9 8791 df-ndx 11967 df-slot 11968 df-base 11970 df-tset 12045 df-rest 12127 df-topn 12128 |
This theorem is referenced by: topnidg 12138 topnpropgd 12139 |
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