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Theorem restsn 17226
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 17055 . . . 4  |-  { (/) }  e.  Top
2 elrest 13647 . . . 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 4331 . . . . 5  |-  (/)  e.  _V
5 ineq1 3527 . . . . . . 7  |-  ( y  =  (/)  ->  ( y  i^i  A )  =  ( (/)  i^i  A ) )
6 incom 3525 . . . . . . . 8  |-  ( (/)  i^i 
A )  =  ( A  i^i  (/) )
7 in0 3645 . . . . . . . 8  |-  ( A  i^i  (/) )  =  (/)
86, 7eqtri 2455 . . . . . . 7  |-  ( (/)  i^i 
A )  =  (/)
95, 8syl6eq 2483 . . . . . 6  |-  ( y  =  (/)  ->  ( y  i^i  A )  =  (/) )
109eqeq2d 2446 . . . . 5  |-  ( y  =  (/)  ->  ( x  =  ( y  i^i 
A )  <->  x  =  (/) ) )
114, 10rexsn 3842 . . . 4  |-  ( E. y  e.  { (/) } x  =  ( y  i^i  A )  <->  x  =  (/) )
12 elsn 3821 . . . 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 2433 1  |-  ( A  e.  V  ->  ( { (/) }t  A )  =  { (/)
} )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   E.wrex 2698    i^i cin 3311   (/)c0 3620   {csn 3806  (class class class)co 6073   ↾t crest 13640   Topctop 16950
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-rest 13642  df-top 16955  df-topon 16958
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