Theorem cldss2 22915

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 22914	. . 3 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ⊆ 𝑋)
3		velpw 4556	. . 3 ⊢ (𝑥 ∈ 𝒫 𝑋 ↔ 𝑥 ⊆ 𝑋)
4	2, 3	sylibr 234	. 2 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ∈ 𝒫 𝑋)
5	4	ssriv 3939	1 ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋

Syntax hints:   = wceq 1540   ∈ wcel 2109   ⊆ wss 3903  𝒫 cpw 4551  ∪ cuni 4858  ‘cfv 6482  Clsdccld 22901

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6438  df-fun 6484  df-fn 6485  df-fv 6490  df-top 22779  df-cld 22904

This theorem is referenced by:  cldmre  22963  cncls2  23158  fclscmp  23915  bcthlem5  25226  ubthlem1  30814  unicls  33870  clsf2  44099