Theorem cldmre 22229

Description: The closed sets of a topology comprise a Moore system on the points of the topology. (Contributed by Stefan O'Rear, 31-Jan-2015.)

Hypothesis

Ref	Expression
clscld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
cldmre	⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘𝑋))

Proof of Theorem cldmre

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clscld.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
2	1	cldss2 22181	. . 3 ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋
3	2	a1i 11	. 2 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ 𝒫 𝑋)
4	1	topcld 22186	. 2 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
5		intcld 22191	. . . 4 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ (Clsd‘𝐽)) → ∩ 𝑥 ∈ (Clsd‘𝐽))
6	5	ancoms 459	. . 3 ⊢ ((𝑥 ⊆ (Clsd‘𝐽) ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ∈ (Clsd‘𝐽))
7	6	3adant1 1129	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ (Clsd‘𝐽) ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ∈ (Clsd‘𝐽))
8	3, 4, 7	ismred 17311	1 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘𝑋))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106   ≠ wne 2943   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  ∪ cuni 4839  ∩ cint 4879  ‘cfv 6433  Moorecmre 17291  Topctop 22042  Clsdccld 22167

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-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-rab 3073  df-v 3434  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-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441  df-mre 17295  df-top 22043  df-cld 22170

This theorem is referenced by:  mrccls  22230  cldmreon  22245  mreclatdemoBAD  22247