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Theorem relexp0d 14171
Description: A relation composed zero times is the (restricted) identity. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)
Hypotheses
Ref Expression
relexp0d.1 (𝜑 → Rel 𝑅)
relexp0d.2 (𝜑𝑅 ∈ V)
Assertion
Ref Expression
relexp0d (𝜑 → (𝑅𝑟0) = ( I ↾ 𝑅))

Proof of Theorem relexp0d
StepHypRef Expression
1 relexp0d.2 . 2 (𝜑𝑅 ∈ V)
2 relexp0d.1 . 2 (𝜑 → Rel 𝑅)
3 relexp0 14170 . 2 ((𝑅 ∈ V ∧ Rel 𝑅) → (𝑅𝑟0) = ( I ↾ 𝑅))
41, 2, 3syl2anc 579 1 (𝜑 → (𝑅𝑟0) = ( I ↾ 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2106  Vcvv 3397   cuni 4671   I cid 5260  cres 5357  Rel wrel 5360  (class class class)co 6922  0cc0 10272  𝑟crelexp 14167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-mulcl 10334  ax-i2m1 10340
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-n0 11643  df-relexp 14168
This theorem is referenced by:  rtrclreclem1  14205  rtrclreclem4  14208
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