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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 6852 | . . . 4 ⊢ ∪ 𝑗 ∈ V | |
2 | 1 | pwex 4774 | . . 3 ⊢ 𝒫 ∪ 𝑗 ∈ V |
3 | 2 | rabex 4740 | . 2 ⊢ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗} ∈ V |
4 | df-cld 20633 | . 2 ⊢ Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗}) | |
5 | 3, 4 | fnmpti 5935 | 1 ⊢ Clsd Fn Top |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 {crab 2900 ∖ cdif 3537 𝒫 cpw 4108 ∪ cuni 4372 Fn wfn 5799 Topctop 20517 Clsdccld 20630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-fun 5806 df-fn 5807 df-cld 20633 |
This theorem is referenced by: cldrcl 20640 iscldtop 20709 |
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