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Theorem cldss2 12576
Description: The set of closed sets is contained in the powerset of the base. (Contributed by Mario Carneiro, 6-Jan-2014.)
Hypothesis
Ref Expression
iscld.1 𝑋 = 𝐽
Assertion
Ref Expression
cldss2 (Clsd‘𝐽) ⊆ 𝒫 𝑋

Proof of Theorem cldss2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 iscld.1 . . . 4 𝑋 = 𝐽
21cldss 12575 . . 3 (𝑥 ∈ (Clsd‘𝐽) → 𝑥𝑋)
3 velpw 3551 . . 3 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
42, 3sylibr 133 . 2 (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ∈ 𝒫 𝑋)
54ssriv 3132 1 (Clsd‘𝐽) ⊆ 𝒫 𝑋
Colors of variables: wff set class
Syntax hints:   = wceq 1335  wcel 2128  wss 3102  𝒫 cpw 3544   cuni 3774  cfv 5172  Clsdccld 12562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171  ax-un 4395
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4028  df-mpt 4029  df-id 4255  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-iota 5137  df-fun 5174  df-fn 5175  df-fv 5180  df-top 12466  df-cld 12565
This theorem is referenced by: (None)
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