Theorem kur14lem5 36096

Description: Lemma for kur14 36102. Closure is an idempotent operation in the set of subsets of a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)

Hypotheses

Ref	Expression
kur14lem.j	⊢ 𝐽 ∈ Top
kur14lem.x	⊢ 𝑋 = ∪ 𝐽
kur14lem.k	⊢ 𝐾 = (cls‘𝐽)
kur14lem.i	⊢ 𝐼 = (int‘𝐽)
kur14lem.a	⊢ 𝐴 ⊆ 𝑋

Assertion

Ref	Expression
kur14lem5	⊢ (𝐾‘(𝐾‘𝐴)) = (𝐾‘𝐴)

Proof of Theorem kur14lem5

Step	Hyp	Ref	Expression
1		kur14lem.j	. . 3 ⊢ 𝐽 ∈ Top
2		kur14lem.a	. . 3 ⊢ 𝐴 ⊆ 𝑋
3		kur14lem.x	. . . 4 ⊢ 𝑋 = ∪ 𝐽
4	3	clsidm 23397	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘((cls‘𝐽)‘𝐴)) = ((cls‘𝐽)‘𝐴))
5	1, 2, 4	mp2an 705	. 2 ⊢ ((cls‘𝐽)‘((cls‘𝐽)‘𝐴)) = ((cls‘𝐽)‘𝐴)
6		kur14lem.k	. . 3 ⊢ 𝐾 = (cls‘𝐽)
7	6	fveq1i 6894	. . 3 ⊢ (𝐾‘𝐴) = ((cls‘𝐽)‘𝐴)
8	6, 7	fveq12i 6899	. 2 ⊢ (𝐾‘(𝐾‘𝐴)) = ((cls‘𝐽)‘((cls‘𝐽)‘𝐴))
9	5, 8, 7	3eqtr4i 2796	1 ⊢ (𝐾‘(𝐾‘𝐴)) = (𝐾‘𝐴)

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2146   ⊆ wss 3899  ∪ cuni 4869  ‘cfv 6546  Topctop 23223  intcnt 23347  clsccl 23348

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-rep 5234  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-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  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-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-top 23224  df-cld 23349  df-cls 23351

This theorem is used by:  kur14lem6  36097  kur14lem7  36098