| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > ntropn | Structured version Visualization version GIF version | ||
| Description: The interior of a subset of a topology's underlying set is open. (Contributed by NM, 11-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
| Ref | Expression |
|---|---|
| clscld.1 | ⊢ 𝑋 = ∪ 𝐽 |
| Ref | Expression |
|---|---|
| ntropn | ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ∈ 𝐽) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | clscld.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
| 2 | 1 | ntrval 23044 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = ∪ (𝐽 ∩ 𝒫 𝑆)) |
| 3 | inss1 4237 | . . . 4 ⊢ (𝐽 ∩ 𝒫 𝑆) ⊆ 𝐽 | |
| 4 | uniopn 22903 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝐽 ∩ 𝒫 𝑆) ⊆ 𝐽) → ∪ (𝐽 ∩ 𝒫 𝑆) ∈ 𝐽) | |
| 5 | 3, 4 | mpan2 691 | . . 3 ⊢ (𝐽 ∈ Top → ∪ (𝐽 ∩ 𝒫 𝑆) ∈ 𝐽) |
| 6 | 5 | adantr 480 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∪ (𝐽 ∩ 𝒫 𝑆) ∈ 𝐽) |
| 7 | 2, 6 | eqeltrd 2841 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ∈ 𝐽) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 ‘cfv 6561 Topctop 22899 intcnt 23025 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-top 22900 df-ntr 23028 |
| This theorem is referenced by: ntrval2 23059 ntrss3 23068 ntrin 23069 cmclsopn 23070 cmntrcld 23071 isopn3 23074 ntridm 23076 neiint 23112 topssnei 23132 maxlp 23155 restntr 23190 iscnp4 23271 cnntri 23279 cnprest 23297 llycmpkgen2 23558 xkococnlem 23667 flimopn 23983 fclsneii 24025 fcfnei 24043 subgntr 24115 iccntr 24843 rectbntr0 24854 bcthlem5 25362 bcth3 25365 limcflf 25916 perfdvf 25938 ubthlem1 30889 cvmlift2lem12 35319 opnregcld 36331 ntrrn 44135 toplatglb 48890 |
| Copyright terms: Public domain | W3C validator |