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Theorem cldss2 12112
 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 12111 . . 3 (𝑥 ∈ (Clsd‘𝐽) → 𝑥𝑋)
3 selpw 3481 . . 3 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
42, 3sylibr 133 . 2 (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ∈ 𝒫 𝑋)
54ssriv 3065 1 (Clsd‘𝐽) ⊆ 𝒫 𝑋
 Colors of variables: wff set class Syntax hints:   = wceq 1312   ∈ wcel 1461   ⊆ wss 3035  𝒫 cpw 3474  ∪ cuni 3700  ‘cfv 5079  Clsdccld 12098 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313 This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-rab 2397  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-iota 5044  df-fun 5081  df-fn 5082  df-fv 5087  df-top 12002  df-cld 12101 This theorem is referenced by: (None)
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