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Theorem kur14lem3 31468
Description: Lemma for kur14 31476. 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 𝐴𝑋
Assertion
Ref Expression
kur14lem3 (𝐾𝐴) ⊆ 𝑋

Proof of Theorem kur14lem3
StepHypRef Expression
1 kur14lem.k . . 3 𝐾 = (cls‘𝐽)
21fveq1i 6341 . 2 (𝐾𝐴) = ((cls‘𝐽)‘𝐴)
3 kur14lem.j . . 3 𝐽 ∈ Top
4 kur14lem.a . . 3 𝐴𝑋
5 kur14lem.x . . . 4 𝑋 = 𝐽
65clsss3 21036 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((cls‘𝐽)‘𝐴) ⊆ 𝑋)
73, 4, 6mp2an 710 . 2 ((cls‘𝐽)‘𝐴) ⊆ 𝑋
82, 7eqsstri 3764 1 (𝐾𝐴) ⊆ 𝑋
Colors of variables: wff setvar class
Syntax hints:   = wceq 1620  wcel 2127  wss 3703   cuni 4576  cfv 6037  Topctop 20871  intcnt 20994  clsccl 20995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-int 4616  df-iun 4662  df-iin 4663  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-top 20872  df-cld 20996  df-cls 20998
This theorem is referenced by:  kur14lem6  31471  kur14lem7  31472
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