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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 11963 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ Top) | |
2 | tsetslid 11891 | . . . . . 6 ⊢ (TopSet = Slot (TopSet‘ndx) ∧ (TopSet‘ndx) ∈ ℕ) | |
3 | 2 | slotslfn 11767 | . . . . 5 ⊢ TopSet Fn V |
4 | fnrel 5157 | . . . . 5 ⊢ (TopSet Fn V → Rel TopSet) | |
5 | 3, 4 | ax-mp 7 | . . . 4 ⊢ Rel TopSet |
6 | 0opn 11955 | . . . . 5 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
7 | tsettps.j | . . . . 5 ⊢ 𝐽 = (TopSet‘𝐾) | |
8 | 6, 7 | syl6eleq 2192 | . . . 4 ⊢ (𝐽 ∈ Top → ∅ ∈ (TopSet‘𝐾)) |
9 | relelfvdm 5385 | . . . 4 ⊢ ((Rel TopSet ∧ ∅ ∈ (TopSet‘𝐾)) → 𝐾 ∈ dom TopSet) | |
10 | 5, 8, 9 | sylancr 408 | . . 3 ⊢ (𝐽 ∈ Top → 𝐾 ∈ dom TopSet) |
11 | 1, 10 | syl 14 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ dom TopSet) |
12 | toponuni 11964 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐴 = ∪ 𝐽) | |
13 | eqimss2 3102 | . . . 4 ⊢ (𝐴 = ∪ 𝐽 → ∪ 𝐽 ⊆ 𝐴) | |
14 | 12, 13 | syl 14 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → ∪ 𝐽 ⊆ 𝐴) |
15 | sspwuni 3843 | . . 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 11915 | . 2 ⊢ ((𝐾 ∈ dom TopSet ∧ 𝐽 ⊆ 𝒫 𝐴) → 𝐽 = (TopOpen‘𝐾)) |
19 | 11, 16, 18 | syl2anc 406 | 1 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopOpen‘𝐾)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1299 ∈ wcel 1448 Vcvv 2641 ⊆ wss 3021 ∅c0 3310 𝒫 cpw 3457 ∪ cuni 3683 dom cdm 4477 Rel wrel 4482 Fn wfn 5054 ‘cfv 5059 Basecbs 11741 TopSetcts 11809 TopOpenctopn 11903 Topctop 11946 TopOnctopon 11959 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-1re 7589 ax-addrcl 7592 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-inn 8579 df-2 8637 df-3 8638 df-4 8639 df-5 8640 df-6 8641 df-7 8642 df-8 8643 df-9 8644 df-ndx 11744 df-slot 11745 df-base 11747 df-tset 11822 df-rest 11904 df-topn 11905 df-top 11947 df-topon 11960 |
This theorem is referenced by: tsettps 11987 |
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