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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 12220 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ Top) | |
2 | tsetslid 12148 | . . . . . 6 ⊢ (TopSet = Slot (TopSet‘ndx) ∧ (TopSet‘ndx) ∈ ℕ) | |
3 | 2 | slotslfn 12024 | . . . . 5 ⊢ TopSet Fn V |
4 | fnrel 5229 | . . . . 5 ⊢ (TopSet Fn V → Rel TopSet) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ Rel TopSet |
6 | 0opn 12212 | . . . . 5 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
7 | tsettps.j | . . . . 5 ⊢ 𝐽 = (TopSet‘𝐾) | |
8 | 6, 7 | eleqtrdi 2233 | . . . 4 ⊢ (𝐽 ∈ Top → ∅ ∈ (TopSet‘𝐾)) |
9 | relelfvdm 5461 | . . . 4 ⊢ ((Rel TopSet ∧ ∅ ∈ (TopSet‘𝐾)) → 𝐾 ∈ dom TopSet) | |
10 | 5, 8, 9 | sylancr 411 | . . 3 ⊢ (𝐽 ∈ Top → 𝐾 ∈ dom TopSet) |
11 | 1, 10 | syl 14 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ dom TopSet) |
12 | toponuni 12221 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐴 = ∪ 𝐽) | |
13 | eqimss2 3157 | . . . 4 ⊢ (𝐴 = ∪ 𝐽 → ∪ 𝐽 ⊆ 𝐴) | |
14 | 12, 13 | syl 14 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝐴) → ∪ 𝐽 ⊆ 𝐴) |
15 | sspwuni 3905 | . . 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 12172 | . 2 ⊢ ((𝐾 ∈ dom TopSet ∧ 𝐽 ⊆ 𝒫 𝐴) → 𝐽 = (TopOpen‘𝐾)) |
19 | 11, 16, 18 | syl2anc 409 | 1 ⊢ (𝐽 ∈ (TopOn‘𝐴) → 𝐽 = (TopOpen‘𝐾)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 Vcvv 2689 ⊆ wss 3076 ∅c0 3368 𝒫 cpw 3515 ∪ cuni 3744 dom cdm 4547 Rel wrel 4552 Fn wfn 5126 ‘cfv 5131 Basecbs 11998 TopSetcts 12066 TopOpenctopn 12160 Topctop 12203 TopOnctopon 12216 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1re 7738 ax-addrcl 7741 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-5 8806 df-6 8807 df-7 8808 df-8 8809 df-9 8810 df-ndx 12001 df-slot 12002 df-base 12004 df-tset 12079 df-rest 12161 df-topn 12162 df-top 12204 df-topon 12217 |
This theorem is referenced by: tsettps 12244 |
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