Theorem cldss2 22089

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.

Step	Hyp	Ref	Expression
1		iscld.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
2	1	cldss 22088	. . 3 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ⊆ 𝑋)
3		velpw 4535	. . 3 ⊢ (𝑥 ∈ 𝒫 𝑋 ↔ 𝑥 ⊆ 𝑋)
4	2, 3	sylibr 233	. 2 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ∈ 𝒫 𝑋)
5	4	ssriv 3921	1 ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋

Syntax hints:   = wceq 1539   ∈ wcel 2108   ⊆ wss 3883  𝒫 cpw 4530  ∪ cuni 4836  ‘cfv 6418  Clsdccld 22075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426  df-top 21951  df-cld 22078

This theorem is referenced by:  cldmre  22137  cncls2  22332  fclscmp  23089  bcthlem5  24397  ubthlem1  29133  unicls  31755  clsf2  41625