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Theorem restsn 18796
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 18625 . . . 4  |-  { (/) }  e.  Top
2 elrest 14387 . . . 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 4443 . . . . 5  |-  (/)  e.  _V
5 ineq1 3566 . . . . . . 7  |-  ( y  =  (/)  ->  ( y  i^i  A )  =  ( (/)  i^i  A ) )
6 incom 3564 . . . . . . . 8  |-  ( (/)  i^i 
A )  =  ( A  i^i  (/) )
7 in0 3684 . . . . . . . 8  |-  ( A  i^i  (/) )  =  (/)
86, 7eqtri 2463 . . . . . . 7  |-  ( (/)  i^i 
A )  =  (/)
95, 8syl6eq 2491 . . . . . 6  |-  ( y  =  (/)  ->  ( y  i^i  A )  =  (/) )
109eqeq2d 2454 . . . . 5  |-  ( y  =  (/)  ->  ( x  =  ( y  i^i 
A )  <->  x  =  (/) ) )
114, 10rexsn 3937 . . . 4  |-  ( E. y  e.  { (/) } x  =  ( y  i^i  A )  <->  x  =  (/) )
12 elsn 3912 . . . 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 2441 1  |-  ( A  e.  V  ->  ( { (/) }t  A )  =  { (/)
} )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756   E.wrex 2737    i^i cin 3348   (/)c0 3658   {csn 3898  (class class class)co 6112   ↾t crest 14380   Topctop 18520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-rest 14382  df-top 18525  df-topon 18528
This theorem is referenced by: (None)
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