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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 2115 | . 2 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | toptopon 12028 | 1 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 1463 ∪ cuni 3702 ‘cfv 5081 Topctop 12007 TopOnctopon 12020 |
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-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-rab 2399 df-v 2659 df-sbc 2879 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-mpt 3951 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-iota 5046 df-fun 5083 df-fv 5089 df-topon 12021 |
This theorem is referenced by: topontopon 12030 lmreltop 12205 cnptopco 12233 cnptopresti 12249 lmtopcnp 12261 lmcn 12262 txcnmpt 12284 txdis1cn 12289 lmcn2 12291 cnmpt1t 12296 cnmpt12 12298 cnmpt21 12302 cnmpt21f 12303 cnmpt2t 12304 cnmpt22 12305 cnmpt22f 12306 cnmptcom 12309 limccnp2lem 12601 limccnp2cntop 12602 |
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