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Theorem lpss3 17209
Description: Subset relationship for limit points. (Contributed by Mario Carneiro, 25-Dec-2016.)
Hypothesis
Ref Expression
lpfval.1  |-  X  = 
U. J
Assertion
Ref Expression
lpss3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  (
( limPt `  J ) `  T )  C_  (
( limPt `  J ) `  S ) )

Proof of Theorem lpss3
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp1 958 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  J  e.  Top )
2 simp2 959 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  S  C_  X )
32ssdifssd 3486 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  ( S  \  { x }
)  C_  X )
4 simp3 960 . . . . . 6  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  T  C_  S )
54ssdifd 3484 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  ( T  \  { x }
)  C_  ( S  \  { x } ) )
6 lpfval.1 . . . . . 6  |-  X  = 
U. J
76clsss 17119 . . . . 5  |-  ( ( J  e.  Top  /\  ( S  \  { x } )  C_  X  /\  ( T  \  {
x } )  C_  ( S  \  { x } ) )  -> 
( ( cls `  J
) `  ( T  \  { x } ) )  C_  ( ( cls `  J ) `  ( S  \  { x } ) ) )
81, 3, 5, 7syl3anc 1185 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  (
( cls `  J
) `  ( T  \  { x } ) )  C_  ( ( cls `  J ) `  ( S  \  { x } ) ) )
98sseld 3348 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  (
x  e.  ( ( cls `  J ) `
 ( T  \  { x } ) )  ->  x  e.  ( ( cls `  J
) `  ( S  \  { x } ) ) ) )
104, 2sstrd 3359 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  T  C_  X )
116islp 17205 . . . 4  |-  ( ( J  e.  Top  /\  T  C_  X )  -> 
( x  e.  ( ( limPt `  J ) `  T )  <->  x  e.  ( ( cls `  J
) `  ( T  \  { x } ) ) ) )
121, 10, 11syl2anc 644 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  (
x  e.  ( (
limPt `  J ) `  T )  <->  x  e.  ( ( cls `  J
) `  ( T  \  { x } ) ) ) )
136islp 17205 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( x  e.  ( ( limPt `  J ) `  S )  <->  x  e.  ( ( cls `  J
) `  ( S  \  { x } ) ) ) )
141, 2, 13syl2anc 644 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  (
x  e.  ( (
limPt `  J ) `  S )  <->  x  e.  ( ( cls `  J
) `  ( S  \  { x } ) ) ) )
159, 12, 143imtr4d 261 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  (
x  e.  ( (
limPt `  J ) `  T )  ->  x  e.  ( ( limPt `  J
) `  S )
) )
1615ssrdv 3355 1  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  (
( limPt `  J ) `  T )  C_  (
( limPt `  J ) `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ w3a 937    = wceq 1653    e. wcel 1726    \ cdif 3318    C_ wss 3321   {csn 3815   U.cuni 4016   ` cfv 5455   Topctop 16959   clsccl 17083   limPtclp 17199
This theorem is referenced by:  perfdvf  19791  lpss2  26461
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-iin 4097  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-top 16964  df-cld 17084  df-cls 17086  df-lp 17201
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