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Mirrors > Home > MPE Home > Th. List > Mathboxes > kur14lem2 | Structured version Visualization version GIF version |
Description: Lemma for kur14 32576. Write interior in terms of closure and complement: 𝑖𝐴 = 𝑐𝑘𝑐𝐴 where 𝑐 is complement and 𝑘 is closure. (Contributed by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
kur14lem.j | ⊢ 𝐽 ∈ Top |
kur14lem.x | ⊢ 𝑋 = ∪ 𝐽 |
kur14lem.k | ⊢ 𝐾 = (cls‘𝐽) |
kur14lem.i | ⊢ 𝐼 = (int‘𝐽) |
kur14lem.a | ⊢ 𝐴 ⊆ 𝑋 |
Ref | Expression |
---|---|
kur14lem2 | ⊢ (𝐼‘𝐴) = (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | kur14lem.j | . . 3 ⊢ 𝐽 ∈ Top | |
2 | kur14lem.a | . . 3 ⊢ 𝐴 ⊆ 𝑋 | |
3 | kur14lem.x | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
4 | 3 | ntrval2 21656 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘𝐴) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝐴)))) |
5 | 1, 2, 4 | mp2an 691 | . 2 ⊢ ((int‘𝐽)‘𝐴) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) |
6 | kur14lem.i | . . 3 ⊢ 𝐼 = (int‘𝐽) | |
7 | 6 | fveq1i 6646 | . 2 ⊢ (𝐼‘𝐴) = ((int‘𝐽)‘𝐴) |
8 | kur14lem.k | . . . 4 ⊢ 𝐾 = (cls‘𝐽) | |
9 | 8 | fveq1i 6646 | . . 3 ⊢ (𝐾‘(𝑋 ∖ 𝐴)) = ((cls‘𝐽)‘(𝑋 ∖ 𝐴)) |
10 | 9 | difeq2i 4047 | . 2 ⊢ (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝐴))) = (𝑋 ∖ ((cls‘𝐽)‘(𝑋 ∖ 𝐴))) |
11 | 5, 7, 10 | 3eqtr4i 2831 | 1 ⊢ (𝐼‘𝐴) = (𝑋 ∖ (𝐾‘(𝑋 ∖ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: = 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: kur14lem6 32571 kur14lem7 32572 |
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