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Mirrors > Home > MPE Home > Th. List > Mathboxes > kur14lem5 | Structured version Visualization version GIF version |
Description: Lemma for kur14 34195. Closure is an idempotent operation in the set of subsets of a topology. (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 |
---|---|
kur14lem5 | ⊢ (𝐾‘(𝐾‘𝐴)) = (𝐾‘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 | clsidm 22562 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘((cls‘𝐽)‘𝐴)) = ((cls‘𝐽)‘𝐴)) |
5 | 1, 2, 4 | mp2an 690 | . 2 ⊢ ((cls‘𝐽)‘((cls‘𝐽)‘𝐴)) = ((cls‘𝐽)‘𝐴) |
6 | kur14lem.k | . . 3 ⊢ 𝐾 = (cls‘𝐽) | |
7 | 6 | fveq1i 6889 | . . 3 ⊢ (𝐾‘𝐴) = ((cls‘𝐽)‘𝐴) |
8 | 6, 7 | fveq12i 6894 | . 2 ⊢ (𝐾‘(𝐾‘𝐴)) = ((cls‘𝐽)‘((cls‘𝐽)‘𝐴)) |
9 | 5, 8, 7 | 3eqtr4i 2770 | 1 ⊢ (𝐾‘(𝐾‘𝐴)) = (𝐾‘𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 ⊆ wss 3947 ∪ cuni 4907 ‘cfv 6540 Topctop 22386 intcnt 22512 clsccl 22513 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-top 22387 df-cld 22514 df-cls 22516 |
This theorem is referenced by: kur14lem6 34190 kur14lem7 34191 |
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