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Mirrors > Home > MPE Home > Th. List > Mathboxes > kur14lem5 | Structured version Visualization version GIF version |
Description: Lemma for kur14 32863. 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 21936 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘((cls‘𝐽)‘𝐴)) = ((cls‘𝐽)‘𝐴)) |
5 | 1, 2, 4 | mp2an 692 | . 2 ⊢ ((cls‘𝐽)‘((cls‘𝐽)‘𝐴)) = ((cls‘𝐽)‘𝐴) |
6 | kur14lem.k | . . 3 ⊢ 𝐾 = (cls‘𝐽) | |
7 | 6 | fveq1i 6707 | . . 3 ⊢ (𝐾‘𝐴) = ((cls‘𝐽)‘𝐴) |
8 | 6, 7 | fveq12i 6712 | . 2 ⊢ (𝐾‘(𝐾‘𝐴)) = ((cls‘𝐽)‘((cls‘𝐽)‘𝐴)) |
9 | 5, 8, 7 | 3eqtr4i 2772 | 1 ⊢ (𝐾‘(𝐾‘𝐴)) = (𝐾‘𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2110 ⊆ wss 3857 ∪ cuni 4809 ‘cfv 6369 Topctop 21762 intcnt 21886 clsccl 21887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-top 21763 df-cld 21888 df-cls 21890 |
This theorem is referenced by: kur14lem6 32858 kur14lem7 32859 |
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