Theorem cldmreon 22979

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 22798	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
2		eqid 2729	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	cldmre 22963	. . 3 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘∪ 𝐽))
4	1, 3	syl 17	. 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → (Clsd‘𝐽) ∈ (Moore‘∪ 𝐽))
5		toponuni 22799	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽)
6	5	fveq2d 6826	. 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 4858  ‘cfv 6482  Moorecmre 17484  Topctop 22778  TopOnctopon 22795  Clsdccld 22901

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 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6438  df-fun 6484  df-fn 6485  df-fv 6490  df-mre 17488  df-top 22779  df-topon 22796  df-cld 22904

This theorem is referenced by:  iscldtop  22980  clduni  48889