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Theorem cldss2 20592
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 20591 . . 3 (𝑥 ∈ (Clsd‘𝐽) → 𝑥𝑋)
3 selpw 4114 . . 3 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
42, 3sylibr 222 . 2 (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ∈ 𝒫 𝑋)
54ssriv 3571 1 (Clsd‘𝐽) ⊆ 𝒫 𝑋
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  wcel 1976  wss 3539  𝒫 cpw 4107   cuni 4366  cfv 5790  Clsdccld 20578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fn 5793  df-fv 5798  df-top 20469  df-cld 20581
This theorem is referenced by:  cldmre  20640  cncls2  20835  fclscmp  21592  bcthlem5  22878  ubthlem1  26944  unicls  29111  clsf2  37268
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