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Theorem 0rest 15327
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 4556 . . . 4  |-  (/)  e.  _V
2 restval 15324 . . . 4  |-  ( (
(/)  e.  _V  /\  A  e.  _V )  ->  ( (/)t  A
)  =  ran  (
x  e.  (/)  |->  ( x  i^i  A ) ) )
31, 2mpan 674 . . 3  |-  ( A  e.  _V  ->  ( (/)t  A
)  =  ran  (
x  e.  (/)  |->  ( x  i^i  A ) ) )
4 mpt0 5723 . . . . 5  |-  ( x  e.  (/)  |->  ( x  i^i 
A ) )  =  (/)
54rneqi 5080 . . . 4  |-  ran  (
x  e.  (/)  |->  ( x  i^i  A ) )  =  ran  (/)
6 rn0 5105 . . . 4  |-  ran  (/)  =  (/)
75, 6eqtri 2451 . . 3  |-  ran  (
x  e.  (/)  |->  ( x  i^i  A ) )  =  (/)
83, 7syl6eq 2479 . 2  |-  ( A  e.  _V  ->  ( (/)t  A
)  =  (/) )
9 relxp 4961 . . . 4  |-  Rel  ( _V  X.  _V )
10 restfn 15322 . . . . . 6  |-t  Fn  ( _V  X.  _V )
11 fndm 5693 . . . . . 6  |-  (t  Fn  ( _V  X.  _V )  ->  domt  =  ( _V  X.  _V ) )
1210, 11ax-mp 5 . . . . 5  |-  domt  =  ( _V  X.  _V )
1312releqi 4937 . . . 4  |-  ( Rel 
domt  <->  Rel  ( _V  X.  _V ) )
149, 13mpbir 212 . . 3  |-  Rel  domt
1514ovprc2 6337 . 2  |-  ( -.  A  e.  _V  ->  (
(/)t  A )  =  (/) )
168, 15pm2.61i 167 1  |-  ( (/)t  A )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    e. wcel 1872   _Vcvv 3080    i^i cin 3435   (/)c0 3761    |-> cmpt 4482    X. cxp 4851   dom cdm 4853   ran crn 4854   Rel wrel 4858    Fn wfn 5596  (class class class)co 6305   ↾t crest 15318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-rest 15320
This theorem is referenced by:  firest  15330  topnval  15332  resstopn  20200  ussval  21272
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