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Mirrors > Home > ILE Home > Th. List > tpspropd | GIF version |
Description: A topological space depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
tpspropd.1 | ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) |
tpspropd.2 | ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿)) |
Ref | Expression |
---|---|
tpspropd | ⊢ (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tpspropd.2 | . . 3 ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿)) | |
2 | tpspropd.1 | . . . 4 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) | |
3 | 2 | fveq2d 5425 | . . 3 ⊢ (𝜑 → (TopOn‘(Base‘𝐾)) = (TopOn‘(Base‘𝐿))) |
4 | 1, 3 | eleq12d 2210 | . 2 ⊢ (𝜑 → ((TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)) ↔ (TopOpen‘𝐿) ∈ (TopOn‘(Base‘𝐿)))) |
5 | eqid 2139 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
6 | eqid 2139 | . . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾) | |
7 | 5, 6 | istps 12209 | . 2 ⊢ (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾))) |
8 | eqid 2139 | . . 3 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
9 | eqid 2139 | . . 3 ⊢ (TopOpen‘𝐿) = (TopOpen‘𝐿) | |
10 | 8, 9 | istps 12209 | . 2 ⊢ (𝐿 ∈ TopSp ↔ (TopOpen‘𝐿) ∈ (TopOn‘(Base‘𝐿))) |
11 | 4, 7, 10 | 3bitr4g 222 | 1 ⊢ (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ‘cfv 5123 Basecbs 11969 TopOpenctopn 12131 TopOnctopon 12187 TopSpctps 12207 |
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-in1 603 ax-in2 604 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 7718 ax-resscn 7719 ax-1re 7721 ax-addrcl 7724 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 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-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 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 8728 df-2 8786 df-3 8787 df-4 8788 df-5 8789 df-6 8790 df-7 8791 df-8 8792 df-9 8793 df-ndx 11972 df-slot 11973 df-base 11975 df-tset 12050 df-rest 12132 df-topn 12133 df-top 12175 df-topon 12188 df-topsp 12208 |
This theorem is referenced by: xmspropd 12656 |
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