Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > topontopon | GIF version |
Description: A topology on a set is a topology on the union of its open sets. (Contributed by BJ, 27-Apr-2021.) |
Ref | Expression |
---|---|
topontopon | ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topontop 12092 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | |
2 | toptopon2 12097 | . 2 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
3 | 1, 2 | sylib 121 | 1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1465 ∪ cuni 3706 ‘cfv 5093 Topctop 12075 TopOnctopon 12088 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-topon 12089 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |