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Theorem restsn 17158
Description: The only subspace topology induced by the topology 
{ (/) }. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
Assertion
Ref Expression
restsn  |-  ( A  e.  V  ->  ( { (/) }t  A )  =  { (/)
} )

Proof of Theorem restsn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sn0top 16988 . . . 4  |-  { (/) }  e.  Top
2 elrest 13584 . . . 4  |-  ( ( { (/) }  e.  Top  /\  A  e.  V )  ->  ( x  e.  ( { (/) }t  A )  <->  E. y  e.  { (/) } x  =  ( y  i^i  A ) ) )
31, 2mpan 652 . . 3  |-  ( A  e.  V  ->  (
x  e.  ( {
(/) }t  A )  <->  E. y  e.  { (/) } x  =  ( y  i^i  A
) ) )
4 0ex 4282 . . . . 5  |-  (/)  e.  _V
5 ineq1 3480 . . . . . . 7  |-  ( y  =  (/)  ->  ( y  i^i  A )  =  ( (/)  i^i  A ) )
6 incom 3478 . . . . . . . 8  |-  ( (/)  i^i 
A )  =  ( A  i^i  (/) )
7 in0 3598 . . . . . . . 8  |-  ( A  i^i  (/) )  =  (/)
86, 7eqtri 2409 . . . . . . 7  |-  ( (/)  i^i 
A )  =  (/)
95, 8syl6eq 2437 . . . . . 6  |-  ( y  =  (/)  ->  ( y  i^i  A )  =  (/) )
109eqeq2d 2400 . . . . 5  |-  ( y  =  (/)  ->  ( x  =  ( y  i^i 
A )  <->  x  =  (/) ) )
114, 10rexsn 3795 . . . 4  |-  ( E. y  e.  { (/) } x  =  ( y  i^i  A )  <->  x  =  (/) )
12 elsn 3774 . . . 4  |-  ( x  e.  { (/) }  <->  x  =  (/) )
1311, 12bitr4i 244 . . 3  |-  ( E. y  e.  { (/) } x  =  ( y  i^i  A )  <->  x  e.  {
(/) } )
143, 13syl6bb 253 . 2  |-  ( A  e.  V  ->  (
x  e.  ( {
(/) }t  A )  <->  x  e.  {
(/) } ) )
1514eqrdv 2387 1  |-  ( A  e.  V  ->  ( { (/) }t  A )  =  { (/)
} )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1717   E.wrex 2652    i^i cin 3264   (/)c0 3573   {csn 3759  (class class class)co 6022   ↾t crest 13577   Topctop 16883
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-rest 13579  df-top 16888  df-topon 16891
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