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Mirrors > Home > MPE Home > Th. List > cldrcl | Structured version Visualization version GIF version |
Description: Reverse closure of the closed set operation. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
Ref | Expression |
---|---|
cldrcl | ⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6924 | . 2 ⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ dom Clsd) | |
2 | fncld 22507 | . . 3 ⊢ Clsd Fn Top | |
3 | 2 | fndmi 6649 | . 2 ⊢ dom Clsd = Top |
4 | 1, 3 | eleqtrdi 2844 | 1 ⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 dom cdm 5674 ‘cfv 6539 Topctop 22376 Clsdccld 22501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-iota 6491 df-fun 6541 df-fn 6542 df-fv 6547 df-cld 22504 |
This theorem is referenced by: cldss 22514 cldopn 22516 difopn 22519 iincld 22524 uncld 22526 cldcls 22527 clsss2 22557 opncldf3 22571 restcldi 22658 restcldr 22659 paste 22779 connsubclo 22909 txcld 23088 cldregopn 35153 clddisj 47437 |
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