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Mirrors > Home > ILE Home > Th. List > toptopon2 | GIF version |
Description: A topology is the same thing as a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.) |
Ref | Expression |
---|---|
toptopon2 | ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2139 | . 2 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | toptopon 12195 | 1 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 1480 ∪ cuni 3736 ‘cfv 5123 Topctop 12174 TopOnctopon 12187 |
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-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-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 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-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 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-iota 5088 df-fun 5125 df-fv 5131 df-topon 12188 |
This theorem is referenced by: topontopon 12197 lmreltop 12372 cnovex 12375 cnptopco 12401 cnptopresti 12417 lmtopcnp 12429 lmcn 12430 txcnmpt 12452 txdis1cn 12457 lmcn2 12459 cnmpt1t 12464 cnmpt12 12466 cnmpt21 12470 cnmpt21f 12471 cnmpt2t 12472 cnmpt22 12473 cnmpt22f 12474 cnmptcom 12477 limccnp2lem 12824 limccnp2cntop 12825 |
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