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Theorem restdis 16925
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 16749 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  Top )
21adantr 451 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A )  ->  ~P A  e.  Top )
3 elpw2g 4190 . . . . 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 16924 . . . 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 2804 . . . . 5  |-  x  e. 
_V
87elpw 3644 . . . 4  |-  ( x  e.  ~P B  <->  x  C_  B
)
9 sstr 3200 . . . . . . . 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 3644 . . . . . 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 2294 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 1632    e. wcel 1696    C_ wss 3165   ~Pcpw 3638  (class class class)co 5874   ↾t crest 13341   Topctop 16647
This theorem is referenced by:  dislly  17239  xkopt  17365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-oadd 6499  df-er 6676  df-en 6880  df-fin 6883  df-fi 7181  df-rest 13343  df-topgen 13360  df-top 16652  df-bases 16654  df-topon 16655
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