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Mirrors > Home > ILE Home > Th. List > fncld | 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 4470 | . . . 4 ⊢ ∪ 𝑗 ∈ V | |
2 | 1 | pwex 4213 | . . 3 ⊢ 𝒫 ∪ 𝑗 ∈ V |
3 | 2 | rabex 4174 | . 2 ⊢ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗} ∈ V |
4 | df-cld 14274 | . 2 ⊢ Clsd = (𝑗 ∈ Top ↦ {𝑥 ∈ 𝒫 ∪ 𝑗 ∣ (∪ 𝑗 ∖ 𝑥) ∈ 𝑗}) | |
5 | 3, 4 | fnmpti 5383 | 1 ⊢ Clsd Fn Top |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2164 {crab 2476 ∖ cdif 3151 𝒫 cpw 3602 ∪ cuni 3836 Fn wfn 5250 Topctop 14176 Clsdccld 14271 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-fun 5257 df-fn 5258 df-cld 14274 |
This theorem is referenced by: cldrcl 14281 |
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