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Theorem relexp0rel 13177
Description: The exponentiation of a class to zero is a relation. (Contributed by RP, 23-May-2020.)
Assertion
Ref Expression
relexp0rel  |-  ( R  e.  V  ->  Rel  ( R ^r 
0 ) )

Proof of Theorem relexp0rel
StepHypRef Expression
1 relres 5138 . 2  |-  Rel  (  _I  |`  ( dom  R  u.  ran  R ) )
2 relexp0g 13162 . . 3  |-  ( R  e.  V  ->  ( R ^r  0 )  =  (  _I  |`  ( dom  R  u.  ran  R
) ) )
32releqd 4924 . 2  |-  ( R  e.  V  ->  ( Rel  ( R ^r 
0 )  <->  Rel  (  _I  |`  ( dom  R  u.  ran  R ) ) ) )
41, 3mpbiri 241 1  |-  ( R  e.  V  ->  Rel  ( R ^r 
0 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1904    u. cun 3388    _I cid 4749   dom cdm 4839   ran crn 4840    |` cres 4841   Rel wrel 4844  (class class class)co 6308   0cc0 9557   ^r crelexp 13160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-mulcl 9619  ax-i2m1 9625
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-n0 10894  df-relexp 13161
This theorem is referenced by:  relexprelg  13178  relexpaddg  13193
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