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Mirrors > Home > MPE Home > Th. List > ntrval2 | Structured version Visualization version GIF version |
Description: Interior expressed in terms of closure. (Contributed by NM, 1-Oct-2007.) |
Ref | Expression |
---|---|
clscld.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
ntrval2 | ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑆)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difss 4059 | . . . . . 6 ⊢ (𝑋 ∖ 𝑆) ⊆ 𝑋 | |
2 | clscld.1 | . . . . . . 7 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | clsval2 21655 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑋 ∖ 𝑆) ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ 𝑆)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝑆))))) |
4 | 1, 3 | mpan2 690 | . . . . 5 ⊢ (𝐽 ∈ Top → ((cls‘𝐽)‘(𝑋 ∖ 𝑆)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝑆))))) |
5 | 4 | adantr 484 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ 𝑆)) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝑆))))) |
6 | dfss4 4185 | . . . . . . . 8 ⊢ (𝑆 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑆)) = 𝑆) | |
7 | 6 | biimpi 219 | . . . . . . 7 ⊢ (𝑆 ⊆ 𝑋 → (𝑋 ∖ (𝑋 ∖ 𝑆)) = 𝑆) |
8 | 7 | fveq2d 6649 | . . . . . 6 ⊢ (𝑆 ⊆ 𝑋 → ((int‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝑆))) = ((int‘𝐽)‘𝑆)) |
9 | 8 | adantl 485 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝑆))) = ((int‘𝐽)‘𝑆)) |
10 | 9 | difeq2d 4050 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ (𝑋 ∖ 𝑆)))) = (𝑋 ∖ ((int‘𝐽)‘𝑆))) |
11 | 5, 10 | eqtrd 2833 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘(𝑋 ∖ 𝑆)) = (𝑋 ∖ ((int‘𝐽)‘𝑆))) |
12 | 11 | difeq2d 4050 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑆))) = (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘𝑆)))) |
13 | 2 | ntropn 21654 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ∈ 𝐽) |
14 | 2 | eltopss 21512 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ ((int‘𝐽)‘𝑆) ∈ 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑋) |
15 | 13, 14 | syldan 594 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) ⊆ 𝑋) |
16 | dfss4 4185 | . . 3 ⊢ (((int‘𝐽)‘𝑆) ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘𝑆))) = ((int‘𝐽)‘𝑆)) | |
17 | 15, 16 | sylib 221 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ (𝑋 ∖ ((int‘𝐽)‘𝑆))) = ((int‘𝐽)‘𝑆)) |
18 | 12, 17 | eqtr2d 2834 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘𝑆) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝑆)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 ⊆ wss 3881 ∪ cuni 4800 ‘cfv 6324 Topctop 21498 intcnt 21622 clsccl 21623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-top 21499 df-cld 21624 df-ntr 21625 df-cls 21626 |
This theorem is referenced by: ntrdif 21657 ntrss 21660 kur14lem2 32567 dssmapntrcls 40831 |
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