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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 2177 | . 2 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | toptopon 13149 | 1 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∈ wcel 2148 ∪ cuni 3807 ‘cfv 5211 Topctop 13128 TopOnctopon 13141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-iota 5173 df-fun 5213 df-fv 5219 df-topon 13142 |
This theorem is referenced by: topontopon 13151 lmreltop 13326 cnovex 13329 cnptopco 13355 cnptopresti 13371 lmtopcnp 13383 lmcn 13384 txcnmpt 13406 txdis1cn 13411 lmcn2 13413 cnmpt1t 13418 cnmpt12 13420 cnmpt21 13424 cnmpt21f 13425 cnmpt2t 13426 cnmpt22 13427 cnmpt22f 13428 cnmptcom 13431 limccnp2lem 13778 limccnp2cntop 13779 |
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