Theorem cldcls 21642

Description: A closed subset equals its own closure. (Contributed by NM, 15-Mar-2007.)

Assertion

Ref	Expression
cldcls	⊢ (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆)

Proof of Theorem cldcls

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cldrcl 21626	. . 3 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2		eqid 2819	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	cldss 21629	. . 3 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ ∪ 𝐽)
4	2	clsval 21637	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
5	1, 3, 4	syl2anc 586	. 2 ⊢ (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥})
6		intmin 4887	. 2 ⊢ (𝑆 ∈ (Clsd‘𝐽) → ∩ {𝑥 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑥} = 𝑆)
7	5, 6	eqtrd 2854	1 ⊢ (𝑆 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑆) = 𝑆)

Syntax hints:   → wi 4   = wceq 1531   ∈ wcel 2108  {crab 3140   ⊆ wss 3934  ∪ cuni 4830  ∩ cint 4867  ‘cfv 6348  Topctop 21493  Clsdccld 21616  clsccl 21618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-top 21494  df-cld 21619  df-cls 21621

This theorem is referenced by:  iscld3  21664  clstop  21669  clsss2  21672  cls0  21680  cncls2  21873  lmcld  21903  fclscmp  22630  metnrmlem1a  23458  lebnumlem1  23557  cmetss  23911  minveclem4  24027  hauseqcn  31131