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Theorem relexp0d 14392
 Description: A relation composed zero times is the (restricted) identity. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.)
Hypotheses
Ref Expression
relexp0d.1 (𝜑 → Rel 𝑅)
relexp0d.2 (𝜑𝑅𝑉)
Assertion
Ref Expression
relexp0d (𝜑 → (𝑅𝑟0) = ( I ↾ 𝑅))

Proof of Theorem relexp0d
StepHypRef Expression
1 relexp0d.2 . 2 (𝜑𝑅𝑉)
2 relexp0d.1 . 2 (𝜑 → Rel 𝑅)
3 relexp0 14391 . 2 ((𝑅𝑉 ∧ Rel 𝑅) → (𝑅𝑟0) = ( I ↾ 𝑅))
41, 2, 3syl2anc 587 1 (𝜑 → (𝑅𝑟0) = ( I ↾ 𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ∪ cuni 4803   I cid 5427   ↾ cres 5524  Rel wrel 5527  (class class class)co 7142  0cc0 10541  ↑𝑟crelexp 14387 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7451  ax-1cn 10599  ax-icn 10600  ax-addcl 10601  ax-mulcl 10603  ax-i2m1 10609 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3722  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4246  df-if 4428  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-iota 6288  df-fun 6331  df-fv 6337  df-ov 7145  df-oprab 7146  df-mpo 7147  df-n0 11901  df-relexp 14388 This theorem is referenced by:  rtrclreclem2  14427  rtrclreclem4  14429
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