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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 7283 | . . . 4 ⊢ ∪ 𝑗 ∈ V | |
2 | 1 | pwex 5131 | . . 3 ⊢ 𝒫 ∪ 𝑗 ∈ V |
3 | 2 | rabex 5088 | . 2 ⊢ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗} ∈ V |
4 | df-cld 21347 | . 2 ⊢ Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗}) | |
5 | 3, 4 | fnmpti 6319 | 1 ⊢ Clsd Fn Top |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2051 {crab 3087 ∖ cdif 3821 𝒫 cpw 4417 ∪ cuni 4709 Fn wfn 6181 Topctop 21221 Clsdccld 21344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ral 3088 df-rex 3089 df-rab 3092 df-v 3412 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-br 4927 df-opab 4989 df-mpt 5006 df-id 5309 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-fun 6188 df-fn 6189 df-cld 21347 |
This theorem is referenced by: cldrcl 21354 iscldtop 21423 |
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