Theorem kur14lem5 35190

Description: Lemma for kur14 35196. 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 22987	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘((cls‘𝐽)‘𝐴)) = ((cls‘𝐽)‘𝐴))
5	1, 2, 4	mp2an 692	. 2 ⊢ ((cls‘𝐽)‘((cls‘𝐽)‘𝐴)) = ((cls‘𝐽)‘𝐴)
6		kur14lem.k	. . 3 ⊢ 𝐾 = (cls‘𝐽)
7	6	fveq1i 6841	. . 3 ⊢ (𝐾‘𝐴) = ((cls‘𝐽)‘𝐴)
8	6, 7	fveq12i 6846	. 2 ⊢ (𝐾‘(𝐾‘𝐴)) = ((cls‘𝐽)‘((cls‘𝐽)‘𝐴))
9	5, 8, 7	3eqtr4i 2762	1 ⊢ (𝐾‘(𝐾‘𝐴)) = (𝐾‘𝐴)

Syntax hints:   = wceq 1540   ∈ wcel 2109   ⊆ wss 3911  ∪ cuni 4867  ‘cfv 6499  Topctop 22813  intcnt 22937  clsccl 22938

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-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-top 22814  df-cld 22939  df-cls 22941

This theorem is referenced by:  kur14lem6  35191  kur14lem7  35192