Theorem clstop 22893

Description: The closure of a topology's underlying set is the entire set. (Contributed by NM, 5-Oct-2007.) (Proof shortened by Jim Kingdon, 11-Mar-2023.)

Hypothesis

Ref	Expression
clscld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
clstop	⊢ (𝐽 ∈ Top → ((cls‘𝐽)‘𝑋) = 𝑋)

Proof of Theorem clstop

Step	Hyp	Ref	Expression
1		clscld.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
2	1	topcld 22859	. 2 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
3		cldcls 22866	. 2 ⊢ (𝑋 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑋) = 𝑋)
4	2, 3	syl 17	1 ⊢ (𝐽 ∈ Top → ((cls‘𝐽)‘𝑋) = 𝑋)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  ∪ cuni 4908  ‘cfv 6543  Topctop 22715  Clsdccld 22840  clsccl 22842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-top 22716  df-cld 22843  df-cls 22845

This theorem is referenced by:  hauscmplem  23230