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Mirrors > Home > ILE Home > Th. List > istpsi | GIF version |
Description: Properties that determine a topological space. (Contributed by NM, 20-Oct-2012.) |
Ref | Expression |
---|---|
istpsi.b | ⊢ (Base‘𝐾) = 𝐴 |
istpsi.j | ⊢ (TopOpen‘𝐾) = 𝐽 |
istpsi.1 | ⊢ 𝐴 = ∪ 𝐽 |
istpsi.2 | ⊢ 𝐽 ∈ Top |
Ref | Expression |
---|---|
istpsi | ⊢ 𝐾 ∈ TopSp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | istpsi.2 | . 2 ⊢ 𝐽 ∈ Top | |
2 | istpsi.1 | . 2 ⊢ 𝐴 = ∪ 𝐽 | |
3 | istpsi.b | . . . 4 ⊢ (Base‘𝐾) = 𝐴 | |
4 | 3 | eqcomi 2144 | . . 3 ⊢ 𝐴 = (Base‘𝐾) |
5 | istpsi.j | . . . 4 ⊢ (TopOpen‘𝐾) = 𝐽 | |
6 | 5 | eqcomi 2144 | . . 3 ⊢ 𝐽 = (TopOpen‘𝐾) |
7 | 4, 6 | istps2 12239 | . 2 ⊢ (𝐾 ∈ TopSp ↔ (𝐽 ∈ Top ∧ 𝐴 = ∪ 𝐽)) |
8 | 1, 2, 7 | mpbir2an 927 | 1 ⊢ 𝐾 ∈ TopSp |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ wcel 1481 ∪ cuni 3744 ‘cfv 5131 Basecbs 11998 TopOpenctopn 12160 Topctop 12203 TopSpctps 12236 |
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-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-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 df-topsp 12237 |
This theorem is referenced by: (None) |
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