Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > toponuni | GIF version |
Description: The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
toponuni | ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | istopon 12183 | . 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = ∪ 𝐽)) | |
2 | 1 | simprbi 273 | 1 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ∪ cuni 3736 ‘cfv 5123 Topctop 12167 TopOnctopon 12180 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-topon 12181 |
This theorem is referenced by: toponunii 12187 toponmax 12195 toponss 12196 toponcom 12197 topgele 12199 topontopn 12207 restuni 12344 resttopon2 12350 lmfval 12364 cnfval 12366 cnpfval 12367 cnprcl2k 12378 ssidcn 12382 iscnp4 12390 cnntr 12397 cncnp 12402 cnptopresti 12410 txtopon 12434 txuni 12435 cnmpt1t 12457 cnmpt2t 12465 cnmpt1res 12468 cnmpt2res 12469 mopnuni 12617 isxms2 12624 limccnp2lem 12817 limccnp2cntop 12818 dvfvalap 12822 dvbss 12826 dvfgg 12829 dvcnp2cntop 12835 dvaddxxbr 12837 dvmulxxbr 12838 |
Copyright terms: Public domain | W3C validator |