Theorem kur14lem3 32686

Description: Lemma for kur14 32694. A closure is a subset of the base set. (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
kur14lem3	⊢ (𝐾‘𝐴) ⊆ 𝑋

Proof of Theorem kur14lem3

Step	Hyp	Ref	Expression
1		kur14lem.k	. . 3 ⊢ 𝐾 = (cls‘𝐽)
2	1	fveq1i 6659	. 2 ⊢ (𝐾‘𝐴) = ((cls‘𝐽)‘𝐴)
3		kur14lem.j	. . 3 ⊢ 𝐽 ∈ Top
4		kur14lem.a	. . 3 ⊢ 𝐴 ⊆ 𝑋
5		kur14lem.x	. . . 4 ⊢ 𝑋 = ∪ 𝐽
6	5	clsss3 21759	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
7	3, 4, 6	mp2an 691	. 2 ⊢ ((cls‘𝐽)‘𝐴) ⊆ 𝑋
8	2, 7	eqsstri 3926	1 ⊢ (𝐾‘𝐴) ⊆ 𝑋

Syntax hints:   = wceq 1538   ∈ wcel 2111   ⊆ wss 3858  ∪ cuni 4798  ‘cfv 6335  Topctop 21593  intcnt 21717  clsccl 21718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-iin 4886  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-top 21594  df-cld 21719  df-cls 21721

This theorem is referenced by:  kur14lem6  32689  kur14lem7  32690