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Theorem restsn 19430
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 19259 . . . 4  |-  { (/) }  e.  Top
2 elrest 14672 . . . 4  |-  ( ( { (/) }  e.  Top  /\  A  e.  V )  ->  ( x  e.  ( { (/) }t  A )  <->  E. y  e.  { (/) } x  =  ( y  i^i  A ) ) )
31, 2mpan 670 . . 3  |-  ( A  e.  V  ->  (
x  e.  ( {
(/) }t  A )  <->  E. y  e.  { (/) } x  =  ( y  i^i  A
) ) )
4 0ex 4570 . . . . 5  |-  (/)  e.  _V
5 ineq1 3686 . . . . . . 7  |-  ( y  =  (/)  ->  ( y  i^i  A )  =  ( (/)  i^i  A ) )
6 incom 3684 . . . . . . . 8  |-  ( (/)  i^i 
A )  =  ( A  i^i  (/) )
7 in0 3804 . . . . . . . 8  |-  ( A  i^i  (/) )  =  (/)
86, 7eqtri 2489 . . . . . . 7  |-  ( (/)  i^i 
A )  =  (/)
95, 8syl6eq 2517 . . . . . 6  |-  ( y  =  (/)  ->  ( y  i^i  A )  =  (/) )
109eqeq2d 2474 . . . . 5  |-  ( y  =  (/)  ->  ( x  =  ( y  i^i 
A )  <->  x  =  (/) ) )
114, 10rexsn 4060 . . . 4  |-  ( E. y  e.  { (/) } x  =  ( y  i^i  A )  <->  x  =  (/) )
12 elsn 4034 . . . 4  |-  ( x  e.  { (/) }  <->  x  =  (/) )
1311, 12bitr4i 252 . . 3  |-  ( E. y  e.  { (/) } x  =  ( y  i^i  A )  <->  x  e.  {
(/) } )
143, 13syl6bb 261 . 2  |-  ( A  e.  V  ->  (
x  e.  ( {
(/) }t  A )  <->  x  e.  {
(/) } ) )
1514eqrdv 2457 1  |-  ( A  e.  V  ->  ( { (/) }t  A )  =  { (/)
} )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1374    e. wcel 1762   E.wrex 2808    i^i cin 3468   (/)c0 3778   {csn 4020  (class class class)co 6275   ↾t crest 14665   Topctop 19154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-rest 14667  df-top 19159  df-topon 19162
This theorem is referenced by: (None)
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