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Mirrors > Home > MPE Home > Th. List > fncld | Structured version Visualization version GIF version |
Description: The closed-set generator is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Ref | Expression |
---|---|
fncld | ⊢ Clsd Fn Top |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vuniex 7459 | . . . 4 ⊢ ∪ 𝑗 ∈ V | |
2 | 1 | pwex 5273 | . . 3 ⊢ 𝒫 ∪ 𝑗 ∈ V |
3 | 2 | rabex 5227 | . 2 ⊢ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗} ∈ V |
4 | df-cld 21621 | . 2 ⊢ Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗}) | |
5 | 3, 4 | fnmpti 6485 | 1 ⊢ Clsd Fn Top |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 {crab 3142 ∖ cdif 3932 𝒫 cpw 4538 ∪ cuni 4831 Fn wfn 6344 Topctop 21495 Clsdccld 21618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-fun 6351 df-fn 6352 df-cld 21621 |
This theorem is referenced by: cldrcl 21628 iscldtop 21697 |
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