Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cldss2 Structured version   Visualization version   GIF version

Theorem cldss2 21056
 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 21055 . . 3 (𝑥 ∈ (Clsd‘𝐽) → 𝑥𝑋)
3 selpw 4309 . . 3 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
42, 3sylibr 224 . 2 (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ∈ 𝒫 𝑋)
54ssriv 3748 1 (Clsd‘𝐽) ⊆ 𝒫 𝑋
 Colors of variables: wff setvar class Syntax hints:   = wceq 1632   ∈ wcel 2139   ⊆ wss 3715  𝒫 cpw 4302  ∪ cuni 4588  ‘cfv 6049  Clsdccld 21042 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057  df-top 20921  df-cld 21045 This theorem is referenced by:  cldmre  21104  cncls2  21299  fclscmp  22055  bcthlem5  23345  ubthlem1  28056  unicls  30279  clsf2  38944
 Copyright terms: Public domain W3C validator