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

Theorem cldss2 22754
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 22753 . . 3 (𝑥 ∈ (Clsd‘𝐽) → 𝑥𝑋)
3 velpw 4606 . . 3 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
42, 3sylibr 233 . 2 (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ∈ 𝒫 𝑋)
54ssriv 3985 1 (Clsd‘𝐽) ⊆ 𝒫 𝑋
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wcel 2104  wss 3947  𝒫 cpw 4601   cuni 4907  cfv 6542  Clsdccld 22740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550  df-top 22616  df-cld 22743
This theorem is referenced by:  cldmre  22802  cncls2  22997  fclscmp  23754  bcthlem5  25076  ubthlem1  30390  unicls  33181  clsf2  43179
  Copyright terms: Public domain W3C validator