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Theorem 0rest 13586
Description: Value of the structure restriction when the topology input is empty. (Contributed by Mario Carneiro, 13-Aug-2015.)
Assertion
Ref Expression
0rest  |-  ( (/)t  A )  =  (/)

Proof of Theorem 0rest
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 0ex 4282 . . . 4  |-  (/)  e.  _V
2 restval 13583 . . . 4  |-  ( (
(/)  e.  _V  /\  A  e.  _V )  ->  ( (/)t  A
)  =  ran  (
x  e.  (/)  |->  ( x  i^i  A ) ) )
31, 2mpan 652 . . 3  |-  ( A  e.  _V  ->  ( (/)t  A
)  =  ran  (
x  e.  (/)  |->  ( x  i^i  A ) ) )
4 mpt0 5514 . . . . 5  |-  ( x  e.  (/)  |->  ( x  i^i 
A ) )  =  (/)
54rneqi 5038 . . . 4  |-  ran  (
x  e.  (/)  |->  ( x  i^i  A ) )  =  ran  (/)
6 rn0 5069 . . . 4  |-  ran  (/)  =  (/)
75, 6eqtri 2409 . . 3  |-  ran  (
x  e.  (/)  |->  ( x  i^i  A ) )  =  (/)
83, 7syl6eq 2437 . 2  |-  ( A  e.  _V  ->  ( (/)t  A
)  =  (/) )
9 relxp 4925 . . . 4  |-  Rel  ( _V  X.  _V )
10 restfn 13581 . . . . . 6  |-t  Fn  ( _V  X.  _V )
11 fndm 5486 . . . . . 6  |-  (t  Fn  ( _V  X.  _V )  ->  domt  =  ( _V  X.  _V ) )
1210, 11ax-mp 8 . . . . 5  |-  domt  =  ( _V  X.  _V )
1312releqi 4902 . . . 4  |-  ( Rel 
domt  <->  Rel  ( _V  X.  _V ) )
149, 13mpbir 201 . . 3  |-  Rel  domt
1514ovprc2 6051 . 2  |-  ( -.  A  e.  _V  ->  (
(/)t  A )  =  (/) )
168, 15pm2.61i 158 1  |-  ( (/)t  A )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717   _Vcvv 2901    i^i cin 3264   (/)c0 3573    e. cmpt 4209    X. cxp 4818   dom cdm 4820   ran crn 4821   Rel wrel 4825    Fn wfn 5391  (class class class)co 6022   ↾t crest 13577
This theorem is referenced by:  firest  13589  topnval  13591  resstopn  17174  ussval  18212
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-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-1st 6290  df-2nd 6291  df-rest 13579
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