ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  restdis Unicode version

Theorem restdis 12259
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 12160 . . . . 5  |-  ( A  e.  V  ->  ~P A  e.  Top )
21adantr 272 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A )  ->  ~P A  e.  Top )
3 elpw2g 4049 . . . . 5  |-  ( A  e.  V  ->  ( B  e.  ~P A  <->  B 
C_  A ) )
43biimpar 293 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A )  ->  B  e.  ~P A
)
5 restopn2 12258 . . . 4  |-  ( ( ~P A  e.  Top  /\  B  e.  ~P A
)  ->  ( x  e.  ( ~P At  B )  <-> 
( x  e.  ~P A  /\  x  C_  B
) ) )
62, 4, 5syl2anc 406 . . 3  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( x  e.  ( ~P At  B )  <->  ( x  e.  ~P A  /\  x  C_  B ) ) )
7 velpw 3485 . . . 4  |-  ( x  e.  ~P B  <->  x  C_  B
)
8 sstr 3073 . . . . . . . 8  |-  ( ( x  C_  B  /\  B  C_  A )  ->  x  C_  A )
98expcom 115 . . . . . . 7  |-  ( B 
C_  A  ->  (
x  C_  B  ->  x 
C_  A ) )
109adantl 273 . . . . . 6  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( x  C_  B  ->  x  C_  A )
)
11 velpw 3485 . . . . . 6  |-  ( x  e.  ~P A  <->  x  C_  A
)
1210, 11syl6ibr 161 . . . . 5  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( x  C_  B  ->  x  e.  ~P A
) )
1312pm4.71rd 389 . . . 4  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( x  C_  B  <->  ( x  e.  ~P A  /\  x  C_  B ) ) )
147, 13syl5bb 191 . . 3  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( x  e.  ~P B 
<->  ( x  e.  ~P A  /\  x  C_  B
) ) )
156, 14bitr4d 190 . 2  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( x  e.  ( ~P At  B )  <->  x  e.  ~P B ) )
1615eqrdv 2113 1  |-  ( ( A  e.  V  /\  B  C_  A )  -> 
( ~P At  B )  =  ~P B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463    C_ wss 3039   ~Pcpw 3478  (class class class)co 5740   ↾t crest 12026   Topctop 12070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-rest 12028  df-topgen 12047  df-top 12071  df-topon 12084  df-bases 12116
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator