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Theorem kur14lem3 28916
Description: Lemma for kur14 28924. A closure is a subset of the base set. (Contributed by Mario Carneiro, 11-Feb-2015.)
Hypotheses
Ref Expression
kur14lem.j  |-  J  e. 
Top
kur14lem.x  |-  X  = 
U. J
kur14lem.k  |-  K  =  ( cls `  J
)
kur14lem.i  |-  I  =  ( int `  J
)
kur14lem.a  |-  A  C_  X
Assertion
Ref Expression
kur14lem3  |-  ( K `
 A )  C_  X

Proof of Theorem kur14lem3
StepHypRef Expression
1 kur14lem.k . . 3  |-  K  =  ( cls `  J
)
21fveq1i 5849 . 2  |-  ( K `
 A )  =  ( ( cls `  J
) `  A )
3 kur14lem.j . . 3  |-  J  e. 
Top
4 kur14lem.a . . 3  |-  A  C_  X
5 kur14lem.x . . . 4  |-  X  = 
U. J
65clsss3 19727 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( ( cls `  J
) `  A )  C_  X )
73, 4, 6mp2an 670 . 2  |-  ( ( cls `  J ) `
 A )  C_  X
82, 7eqsstri 3519 1  |-  ( K `
 A )  C_  X
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398    e. wcel 1823    C_ wss 3461   U.cuni 4235   ` cfv 5570   Topctop 19561   intcnt 19685   clsccl 19686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-top 19566  df-cld 19687  df-cls 19689
This theorem is referenced by:  kur14lem6  28919  kur14lem7  28920
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