Theorem mrccls 22999

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 2729	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	clsfval 22945	. 2 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = (𝑎 ∈ 𝒫 ∪ 𝐽 ↦ ∩ {𝑏 ∈ (Clsd‘𝐽) ∣ 𝑎 ⊆ 𝑏}))
3	1	cldmre 22998	. . 3 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘∪ 𝐽))
4		mrccls.f	. . . 4 ⊢ 𝐹 = (mrCls‘(Clsd‘𝐽))
5	4	mrcfval 17549	. . 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 2767	1 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = 𝐹)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {crab 3402   ⊆ wss 3911  𝒫 cpw 4559  ∪ cuni 4867  ∩ cint 4906   ↦ cmpt 5183  ‘cfv 6499  Moorecmre 17519  mrClscmrc 17520  Topctop 22813  Clsdccld 22936  clsccl 22938

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-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-mre 17523  df-mrc 17524  df-top 22814  df-cld 22939  df-cls 22941

This theorem is referenced by:  istopclsd  42681