| Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 2204 | . 2 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
| 2 | 1 | toptopon 14432 | 1 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) |
| Colors of variables: wff set class |
| Syntax hints: ↔ wb 105 ∈ wcel 2175 ∪ cuni 3849 ‘cfv 5270 Topctop 14411 TopOnctopon 14424 |
| 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 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252 ax-un 4479 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1375 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ral 2488 df-rex 2489 df-rab 2492 df-v 2773 df-sbc 2998 df-un 3169 df-in 3171 df-ss 3178 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-br 4044 df-opab 4105 df-mpt 4106 df-id 4339 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-iota 5231 df-fun 5272 df-fv 5278 df-topon 14425 |
| This theorem is referenced by: topontopon 14434 lmreltop 14607 cnovex 14610 cnptopco 14636 cnptopresti 14652 lmtopcnp 14664 lmcn 14665 txcnmpt 14687 txdis1cn 14692 lmcn2 14694 cnmpt1t 14699 cnmpt12 14701 cnmpt21 14705 cnmpt21f 14706 cnmpt2t 14707 cnmpt22 14708 cnmpt22f 14709 cnmptcom 14712 limccnp2lem 15090 limccnp2cntop 15091 |
| Copyright terms: Public domain | W3C validator |