Theorem kur14lem3 32568

Description: Lemma for kur14 32576. 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 6646	. 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 21664	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
7	3, 4, 6	mp2an 691	. 2 ⊢ ((cls‘𝐽)‘𝐴) ⊆ 𝑋
8	2, 7	eqsstri 3949	1 ⊢ (𝐾‘𝐴) ⊆ 𝑋

Syntax hints:   = wceq 1538   ∈ wcel 2111   ⊆ wss 3881  ∪ cuni 4800  ‘cfv 6324  Topctop 21498  intcnt 21622  clsccl 21623

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 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-top 21499  df-cld 21624  df-cls 21626

This theorem is referenced by:  kur14lem6  32571  kur14lem7  32572