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Mirrors > Home > ILE Home > Th. List > istps2 | GIF version |
Description: Express the predicate "is a topological space." (Contributed by NM, 20-Oct-2012.) |
Ref | Expression |
---|---|
istps.a | ⊢ 𝐴 = (Base‘𝐾) |
istps.j | ⊢ 𝐽 = (TopOpen‘𝐾) |
Ref | Expression |
---|---|
istps2 | ⊢ (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = ∪ 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | istps.a | . . 3 ⊢ 𝐴 = (Base‘𝐾) | |
2 | istps.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝐾) | |
3 | 1, 2 | istps 12472 | . 2 ⊢ (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐴)) |
4 | istopon 12453 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐴) ↔ (𝐽 ∈ Top ∧ 𝐴 = ∪ 𝐽)) | |
5 | 3, 4 | bitri 183 | 1 ⊢ (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = ∪ 𝐽)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1335 ∈ wcel 2128 ∪ cuni 3773 ‘cfv 5171 Basecbs 12232 TopOpenctopn 12394 Topctop 12437 TopOnctopon 12450 TopSpctps 12470 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4080 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-cnex 7824 ax-resscn 7825 ax-1re 7827 ax-addrcl 7830 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-inn 8835 df-2 8893 df-3 8894 df-4 8895 df-5 8896 df-6 8897 df-7 8898 df-8 8899 df-9 8900 df-ndx 12235 df-slot 12236 df-base 12238 df-tset 12313 df-rest 12395 df-topn 12396 df-top 12438 df-topon 12451 df-topsp 12471 |
This theorem is referenced by: tpsuni 12474 tpstop 12475 istpsi 12479 |
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