Theorem cldmreon 22981

Description: The closed sets of a topology over a set are a Moore collection over the same set. (Contributed by Stefan O'Rear, 31-Jan-2015.)

Assertion

Ref	Expression
cldmreon	⊢ (𝐽 ∈ (TopOn‘𝐵) → (Clsd‘𝐽) ∈ (Moore‘𝐵))

Proof of Theorem cldmreon

Step	Hyp	Ref	Expression
1		topontop 22800	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
2		eqid 2729	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	cldmre 22965	. . 3 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘∪ 𝐽))
4	1, 3	syl 17	. 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → (Clsd‘𝐽) ∈ (Moore‘∪ 𝐽))
5		toponuni 22801	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽)
6	5	fveq2d 6862	. 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → (Moore‘𝐵) = (Moore‘∪ 𝐽))
7	4, 6	eleqtrrd 2831	1 ⊢ (𝐽 ∈ (TopOn‘𝐵) → (Clsd‘𝐽) ∈ (Moore‘𝐵))

Syntax hints:   → wi 4   ∈ wcel 2109  ∪ cuni 4871  ‘cfv 6511  Moorecmre 17543  Topctop 22780  TopOnctopon 22797  Clsdccld 22903

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-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fn 6514  df-fv 6519  df-mre 17547  df-top 22781  df-topon 22798  df-cld 22906

This theorem is referenced by:  iscldtop  22982  clduni  48889