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Theorem relexp0rel 14384
Description: The exponentiation of a class to zero is a relation. (Contributed by RP, 23-May-2020.)
Assertion
Ref Expression
relexp0rel (𝑅𝑉 → Rel (𝑅𝑟0))

Proof of Theorem relexp0rel
StepHypRef Expression
1 relres 5875 . 2 Rel ( I ↾ (dom 𝑅 ∪ ran 𝑅))
2 relexp0g 14369 . . 3 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
32releqd 5646 . 2 (𝑅𝑉 → (Rel (𝑅𝑟0) ↔ Rel ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
41, 3mpbiri 259 1 (𝑅𝑉 → Rel (𝑅𝑟0))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  cun 3931   I cid 5452  dom cdm 5548  ran crn 5549  cres 5550  Rel wrel 5553  (class class class)co 7145  0cc0 10525  𝑟crelexp 14367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-mulcl 10587  ax-i2m1 10593
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-n0 11886  df-relexp 14368
This theorem is referenced by:  relexprelg  14385  relexpaddg  14400
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