Theorem cldss2 14426

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 14425	. . 3 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ⊆ 𝑋)
3		velpw 3613	. . 3 ⊢ (𝑥 ∈ 𝒫 𝑋 ↔ 𝑥 ⊆ 𝑋)
4	2, 3	sylibr 134	. 2 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ∈ 𝒫 𝑋)
5	4	ssriv 3188	1 ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋

Syntax hints:   = wceq 1364   ∈ wcel 2167   ⊆ wss 3157  𝒫 cpw 3606  ∪ cuni 3840  ‘cfv 5259  Clsdccld 14412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-top 14318  df-cld 14415

This theorem is referenced by: (None)