Theorem cldmre 20882

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 20834	. . 3 ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋
3	2	a1i 11	. 2 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ⊆ 𝒫 𝑋)
4	1	topcld 20839	. 2 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
5		intcld 20844	. . . 4 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ (Clsd‘𝐽)) → ∩ 𝑥 ∈ (Clsd‘𝐽))
6	5	ancoms 469	. . 3 ⊢ ((𝑥 ⊆ (Clsd‘𝐽) ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ∈ (Clsd‘𝐽))
7	6	3adant1 1079	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ (Clsd‘𝐽) ∧ 𝑥 ≠ ∅) → ∩ 𝑥 ∈ (Clsd‘𝐽))
8	3, 4, 7	ismred 16262	1 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘𝑋))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990   ≠ wne 2794   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  ∪ cuni 4436  ∩ cint 4475  ‘cfv 5888  Moorecmre 16242  Topctop 20698  Clsdccld 20820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-mre 16246  df-top 20699  df-cld 20823

This theorem is referenced by:  mrccls  20883  cldmreon  20898  mreclatdemoBAD  20900