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Theorem restsn 19856
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 19684 . . . 4  |-  { (/) }  e.  Top
2 elrest 14934 . . . 4  |-  ( ( { (/) }  e.  Top  /\  A  e.  V )  ->  ( x  e.  ( { (/) }t  A )  <->  E. y  e.  { (/) } x  =  ( y  i^i  A ) ) )
31, 2mpan 668 . . 3  |-  ( A  e.  V  ->  (
x  e.  ( {
(/) }t  A )  <->  E. y  e.  { (/) } x  =  ( y  i^i  A
) ) )
4 0ex 4525 . . . . 5  |-  (/)  e.  _V
5 ineq1 3633 . . . . . . 7  |-  ( y  =  (/)  ->  ( y  i^i  A )  =  ( (/)  i^i  A ) )
6 incom 3631 . . . . . . . 8  |-  ( (/)  i^i 
A )  =  ( A  i^i  (/) )
7 in0 3764 . . . . . . . 8  |-  ( A  i^i  (/) )  =  (/)
86, 7eqtri 2431 . . . . . . 7  |-  ( (/)  i^i 
A )  =  (/)
95, 8syl6eq 2459 . . . . . 6  |-  ( y  =  (/)  ->  ( y  i^i  A )  =  (/) )
109eqeq2d 2416 . . . . 5  |-  ( y  =  (/)  ->  ( x  =  ( y  i^i 
A )  <->  x  =  (/) ) )
114, 10rexsn 4011 . . . 4  |-  ( E. y  e.  { (/) } x  =  ( y  i^i  A )  <->  x  =  (/) )
12 elsn 3985 . . . 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 2399 1  |-  ( A  e.  V  ->  ( { (/) }t  A )  =  { (/)
} )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1405    e. wcel 1842   E.wrex 2754    i^i cin 3412   (/)c0 3737   {csn 3971  (class class class)co 6234   ↾t crest 14927   Topctop 19578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-rest 14929  df-top 19583  df-topon 19586
This theorem is referenced by: (None)
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