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Theorem restdis 16909
Description: A subspace of a discrete topology is discrete. (Contributed by Mario Carneiro, 19-Mar-2015.)
Assertion
Ref Expression
restdis  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( ~P At  B )  =  ~P B )

Proof of Theorem restdis
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 distop 16733 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  Top )
21adantr 451 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A )  ->  ~P A  e.  Top )
3 elpw2g 4174 . . . . 5  |-  ( A  e.  V  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
43biimpar 471 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A )  ->  B  e.  ~P A
)
5 restopn2 16908 . . . 4  |-  ( ( ~P A  e.  Top  /\  B  e.  ~P A
)  ->  ( x  e.  ( ~P At  B )  <-> 
( x  e.  ~P A  /\  x  C_  B
) ) )
62, 4, 5syl2anc 642 . . 3  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( x  e.  ( ~P At  B )  <->  ( x  e.  ~P A  /\  x  C_  B ) ) )
7 vex 2791 . . . . 5  |-  x  e. 
_V
87elpw 3631 . . . 4  |-  ( x  e.  ~P B  <->  x  C_  B
)
9 sstr 3187 . . . . . . . 8  |-  ( ( x  C_  B  /\  B  C_  A )  ->  x  C_  A )
109expcom 424 . . . . . . 7  |-  ( B 
C_  A  ->  (
x  C_  B  ->  x 
C_  A ) )
1110adantl 452 . . . . . 6  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( x  C_  B  ->  x  C_  A )
)
127elpw 3631 . . . . . 6  |-  ( x  e.  ~P A  <->  x  C_  A
)
1311, 12syl6ibr 218 . . . . 5  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( x  C_  B  ->  x  e.  ~P A
) )
1413pm4.71rd 616 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( x  C_  B  <->  ( x  e.  ~P A  /\  x  C_  B ) ) )
158, 14syl5bb 248 . . 3  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( x  e.  ~P B 
<->  ( x  e.  ~P A  /\  x  C_  B
) ) )
166, 15bitr4d 247 . 2  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( x  e.  ( ~P At  B )  <->  x  e.  ~P B ) )
1716eqrdv 2281 1  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( ~P At  B )  =  ~P B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   ~Pcpw 3625  (class class class)co 5858   ↾t crest 13325   Topctop 16631
This theorem is referenced by:  dislly  17223  xkopt  17349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-oadd 6483  df-er 6660  df-en 6864  df-fin 6867  df-fi 7165  df-rest 13327  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639
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