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Theorem relexp0d 15003
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 15002 . 2 ((𝑅𝑉 ∧ Rel 𝑅) → (𝑅𝑟0) = ( I ↾ 𝑅))
41, 2, 3syl2anc 583 1 (𝜑 → (𝑅𝑟0) = ( I ↾ 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099   cuni 4908   I cid 5575  cres 5680  Rel wrel 5683  (class class class)co 7420  0cc0 11138  𝑟crelexp 14998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-mulcl 11200  ax-i2m1 11206
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-n0 12503  df-relexp 14999
This theorem is referenced by:  rtrclreclem2  15038  rtrclreclem4  15040
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