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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 12181 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ Top) | |
2 | tsetslid 12109 | . . . . . 6 ⊢ (TopSet = Slot (TopSet‘ndx) ∧ (TopSet‘ndx) ∈ ℕ) | |
3 | 2 | slotslfn 11985 | . . . . 5 ⊢ TopSet Fn V |
4 | fnrel 5221 | . . . . 5 ⊢ (TopSet Fn V → Rel TopSet) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ Rel TopSet |
6 | 0opn 12173 | . . . . 5 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
7 | tsettps.j | . . . . 5 ⊢ 𝐽 = (TopSet‘𝐾) | |
8 | 6, 7 | eleqtrdi 2232 | . . . 4 ⊢ (𝐽 ∈ Top → ∅ ∈ (TopSet‘𝐾)) |
9 | relelfvdm 5453 | . . . 4 ⊢ ((Rel TopSet ∧ ∅ ∈ (TopSet‘𝐾)) → 𝐾 ∈ dom TopSet) | |
10 | 5, 8, 9 | sylancr 410 | . . 3 ⊢ (𝐽 ∈ Top → 𝐾 ∈ dom TopSet) |
11 | 1, 10 | syl 14 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ dom TopSet) |
12 | toponuni 12182 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐴 = ∪ 𝐽) | |
13 | eqimss2 3152 | . . . 4 ⊢ (𝐴 = ∪ 𝐽 → ∪ 𝐽 ⊆ 𝐴) | |
14 | 12, 13 | syl 14 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → ∪ 𝐽 ⊆ 𝐴) |
15 | sspwuni 3897 | . . 3 ⊢ (𝐽 ⊆ 𝒫 𝐴 ↔ ∪ 𝐽 ⊆ 𝐴) | |
16 | 14, 15 | sylibr 133 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ⊆ 𝒫 𝐴) |
17 | tsettps.a | . . 3 ⊢ 𝐴 = (Base‘𝐾) | |
18 | 17, 7 | topnidg 12133 | . 2 ⊢ ((𝐾 ∈ dom TopSet ∧ 𝐽 ⊆ 𝒫 𝐴) → 𝐽 = (TopOpen‘𝐾)) |
19 | 11, 16, 18 | syl2anc 408 | 1 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopOpen‘𝐾)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ⊆ wss 3071 ∅c0 3363 𝒫 cpw 3510 ∪ cuni 3736 dom cdm 4539 Rel wrel 4544 Fn wfn 5118 ‘cfv 5123 Basecbs 11959 TopSetcts 12027 TopOpenctopn 12121 Topctop 12164 TopOnctopon 12177 |
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-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1re 7714 ax-addrcl 7717 |
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-ne 2309 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 8721 df-2 8779 df-3 8780 df-4 8781 df-5 8782 df-6 8783 df-7 8784 df-8 8785 df-9 8786 df-ndx 11962 df-slot 11963 df-base 11965 df-tset 12040 df-rest 12122 df-topn 12123 df-top 12165 df-topon 12178 |
This theorem is referenced by: tsettps 12205 |
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