Theorem mrccls 21097

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 2771	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	clsfval 21043	. 2 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = (𝑎 ∈ 𝒫 ∪ 𝐽 ↦ ∩ {𝑏 ∈ (Clsd‘𝐽) ∣ 𝑎 ⊆ 𝑏}))
3	1	cldmre 21096	. . 3 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘∪ 𝐽))
4		mrccls.f	. . . 4 ⊢ 𝐹 = (mrCls‘(Clsd‘𝐽))
5	4	mrcfval 16469	. . 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 2808	1 ⊢ (𝐽 ∈ Top → (cls‘𝐽) = 𝐹)

Syntax hints:   → wi 4   = wceq 1631   ∈ wcel 2145  {crab 3065   ⊆ wss 3723  𝒫 cpw 4297  ∪ cuni 4574  ∩ cint 4611   ↦ cmpt 4863  ‘cfv 6029  Moorecmre 16443  mrClscmrc 16444  Topctop 20911  Clsdccld 21034  clsccl 21036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-mre 16447  df-mrc 16448  df-top 20912  df-cld 21037  df-cls 21039

This theorem is referenced by:  istopclsd  37782