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Theorem 0rest 13647
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 4331 . . . 4  |-  (/)  e.  _V
2 restval 13644 . . . 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 5564 . . . . 5  |-  ( x  e.  (/)  |->  ( x  i^i 
A ) )  =  (/)
54rneqi 5088 . . . 4  |-  ran  (
x  e.  (/)  |->  ( x  i^i  A ) )  =  ran  (/)
6 rn0 5119 . . . 4  |-  ran  (/)  =  (/)
75, 6eqtri 2455 . . 3  |-  ran  (
x  e.  (/)  |->  ( x  i^i  A ) )  =  (/)
83, 7syl6eq 2483 . 2  |-  ( A  e.  _V  ->  ( (/)t  A
)  =  (/) )
9 relxp 4975 . . . 4  |-  Rel  ( _V  X.  _V )
10 restfn 13642 . . . . . 6  |-t  Fn  ( _V  X.  _V )
11 fndm 5536 . . . . . 6  |-  (t  Fn  ( _V  X.  _V )  ->  domt  =  ( _V  X.  _V ) )
1210, 11ax-mp 8 . . . . 5  |-  domt  =  ( _V  X.  _V )
1312releqi 4952 . . . 4  |-  ( Rel 
domt  <->  Rel  ( _V  X.  _V ) )
149, 13mpbir 201 . . 3  |-  Rel  domt
1514ovprc2 6102 . 2  |-  ( -.  A  e.  _V  ->  (
(/)t  A )  =  (/) )
168, 15pm2.61i 158 1  |-  ( (/)t  A )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   _Vcvv 2948    i^i cin 3311   (/)c0 3620    e. cmpt 4258    X. cxp 4868   dom cdm 4870   ran crn 4871   Rel wrel 4875    Fn wfn 5441  (class class class)co 6073   ↾t crest 13638
This theorem is referenced by:  firest  13650  topnval  13652  resstopn  17240  ussval  18279
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-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-1st 6341  df-2nd 6342  df-rest 13640
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