Theorem cldmreon 23424

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 23243	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
2		eqid 2763	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	cldmre 23408	. . 3 ⊢ (𝐽 ∈ Top → (Clsd‘𝐽) ∈ (Moore‘∪ 𝐽))
4	1, 3	syl 18	. 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → (Clsd‘𝐽) ∈ (Moore‘∪ 𝐽))
5		toponuni 23244	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽)
6	5	fveq2d 6897	. 2 ⊢ (𝐽 ∈ (TopOn‘𝐵) → (Moore‘𝐵) = (Moore‘∪ 𝐽))
7	4, 6	eleqtrrd 2866	1 ⊢ (𝐽 ∈ (TopOn‘𝐵) → (Clsd‘𝐽) ∈ (Moore‘𝐵))

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  ∪ cuni 4869  ‘cfv 6546  Moorecmre 17764  Topctop 23223  TopOnctopon 23240  Clsdccld 23346

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-int 4910  df-iun 4955  df-iin 4956  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6500  df-fun 6548  df-fn 6549  df-fv 6554  df-mre 17768  df-top 23224  df-topon 23241  df-cld 23349

This theorem is used by:  iscldtop  23425  clduni  50154