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Mirrors > Home > ILE Home > Th. List > topontopn | GIF version |
Description: Express the predicate "is a topological space". (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
tsettps.a | ⊢ 𝐴 = (Base‘𝐾) |
tsettps.j | ⊢ 𝐽 = (TopSet‘𝐾) |
Ref | Expression |
---|---|
topontopn | ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopOpen‘𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topontop 13966 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ Top) | |
2 | tsetslid 12696 | . . . . . 6 ⊢ (TopSet = Slot (TopSet‘ndx) ∧ (TopSet‘ndx) ∈ ℕ) | |
3 | 2 | slotslfn 12537 | . . . . 5 ⊢ TopSet Fn V |
4 | fnrel 5333 | . . . . 5 ⊢ (TopSet Fn V → Rel TopSet) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ Rel TopSet |
6 | 0opn 13958 | . . . . 5 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
7 | tsettps.j | . . . . 5 ⊢ 𝐽 = (TopSet‘𝐾) | |
8 | 6, 7 | eleqtrdi 2282 | . . . 4 ⊢ (𝐽 ∈ Top → ∅ ∈ (TopSet‘𝐾)) |
9 | relelfvdm 5566 | . . . 4 ⊢ ((Rel TopSet ∧ ∅ ∈ (TopSet‘𝐾)) → 𝐾 ∈ dom TopSet) | |
10 | 5, 8, 9 | sylancr 414 | . . 3 ⊢ (𝐽 ∈ Top → 𝐾 ∈ dom TopSet) |
11 | 1, 10 | syl 14 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ dom TopSet) |
12 | toponuni 13967 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐴 = ∪ 𝐽) | |
13 | eqimss2 3225 | . . . 4 ⊢ (𝐴 = ∪ 𝐽 → ∪ 𝐽 ⊆ 𝐴) | |
14 | 12, 13 | syl 14 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → ∪ 𝐽 ⊆ 𝐴) |
15 | sspwuni 3986 | . . 3 ⊢ (𝐽 ⊆ 𝒫 𝐴 ↔ ∪ 𝐽 ⊆ 𝐴) | |
16 | 14, 15 | sylibr 134 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ⊆ 𝒫 𝐴) |
17 | tsettps.a | . . 3 ⊢ 𝐴 = (Base‘𝐾) | |
18 | 17, 7 | topnidg 12754 | . 2 ⊢ ((𝐾 ∈ dom TopSet ∧ 𝐽 ⊆ 𝒫 𝐴) → 𝐽 = (TopOpen‘𝐾)) |
19 | 11, 16, 18 | syl2anc 411 | 1 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopOpen‘𝐾)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 Vcvv 2752 ⊆ wss 3144 ∅c0 3437 𝒫 cpw 3590 ∪ cuni 3824 dom cdm 4644 Rel wrel 4649 Fn wfn 5230 ‘cfv 5235 Basecbs 12511 TopSetcts 12592 TopOpenctopn 12742 Topctop 13949 TopOnctopon 13962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 ax-1re 7934 ax-addrcl 7937 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-ov 5898 df-oprab 5899 df-mpo 5900 df-1st 6164 df-2nd 6165 df-inn 8949 df-2 9007 df-3 9008 df-4 9009 df-5 9010 df-6 9011 df-7 9012 df-8 9013 df-9 9014 df-ndx 12514 df-slot 12515 df-base 12517 df-tset 12605 df-rest 12743 df-topn 12744 df-top 13950 df-topon 13963 |
This theorem is referenced by: tsettps 13990 |
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