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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 2189 | . 2 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | toptopon 13995 | 1 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∈ wcel 2160 ∪ cuni 3824 ‘cfv 5235 Topctop 13974 TopOnctopon 13987 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-topon 13988 |
This theorem is referenced by: topontopon 13997 lmreltop 14170 cnovex 14173 cnptopco 14199 cnptopresti 14215 lmtopcnp 14227 lmcn 14228 txcnmpt 14250 txdis1cn 14255 lmcn2 14257 cnmpt1t 14262 cnmpt12 14264 cnmpt21 14268 cnmpt21f 14269 cnmpt2t 14270 cnmpt22 14271 cnmpt22f 14272 cnmptcom 14275 limccnp2lem 14622 limccnp2cntop 14623 |
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