Theorem mrccls 22230

Description: Moore closure generalizes closure in a topology. (Contributed by Stefan O'Rear, 31-Jan-2015.)

Hypothesis

Ref	Expression
mrccls.f	⊢ 𝐹 = (mrCls‘(Clsd‘𝐽))

Assertion

Ref	Expression
mrccls	⊢ (𝐽 ∈ Top → (cls‘𝐽) = 𝐹)

Proof of Theorem mrccls

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2738	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	clsfval 22176	. 2 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = (𝑎 ∈ 𝒫 ∪ 𝐽 ↦ ∩ {𝑏 ∈ (Clsd‘𝐽) ∣ 𝑎 ⊆ 𝑏}))
3	1	cldmre 22229	. . 3 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘∪ 𝐽))
4		mrccls.f	. . . 4 ⊢ 𝐹 = (mrCls‘(Clsd‘𝐽))
5	4	mrcfval 17317	. . 3 ⊢ ((Clsd‘𝐽) ∈ (Moore‘∪ 𝐽) → 𝐹 = (𝑎 ∈ 𝒫 ∪ 𝐽 ↦ ∩ {𝑏 ∈ (Clsd‘𝐽) ∣ 𝑎 ⊆ 𝑏}))
6	3, 5	syl 17	. 2 ⊢ (𝐽 ∈ Top → 𝐹 = (𝑎 ∈ 𝒫 ∪ 𝐽 ↦ ∩ {𝑏 ∈ (Clsd‘𝐽) ∣ 𝑎 ⊆ 𝑏}))
7	2, 6	eqtr4d 2781	1 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = 𝐹)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  {crab 3068   ⊆ wss 3887  𝒫 cpw 4533  ∪ cuni 4839  ∩ cint 4879   ↦ cmpt 5157  ‘cfv 6433  Moorecmre 17291  mrClscmrc 17292  Topctop 22042  Clsdccld 22167  clsccl 22169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-mre 17295  df-mrc 17296  df-top 22043  df-cld 22170  df-cls 22172

This theorem is referenced by:  istopclsd  40522