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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 6130 | . 2 ⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ dom Clsd) | |
2 | fncld 20636 | . . 3 ⊢ Clsd Fn Top | |
3 | fndm 5904 | . . 3 ⊢ (Clsd Fn Top → dom Clsd = Top) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ dom Clsd = Top |
5 | 1, 4 | syl6eleq 2698 | 1 ⊢ (𝐶 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 dom cdm 5038 Fn wfn 5799 ‘cfv 5804 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-ne 2782 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-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 df-cld 20633 |
This theorem is referenced by: cldss 20643 cldopn 20645 difopn 20648 iincld 20653 uncld 20655 cldcls 20656 clsss2 20686 opncldf3 20700 restcldi 20787 restcldr 20788 paste 20908 consubclo 21037 txcld 21216 cldregopn 31496 |
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