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Theorem relexp0d 14918
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 14917 . 2 ((𝑅𝑉 ∧ Rel 𝑅) → (𝑅𝑟0) = ( I ↾ 𝑅))
41, 2, 3syl2anc 585 1 (𝜑 → (𝑅𝑟0) = ( I ↾ 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107   cuni 4869   I cid 5534  cres 5639  Rel wrel 5642  (class class class)co 7361  0cc0 11059  𝑟crelexp 14913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-mulcl 11121  ax-i2m1 11127
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-n0 12422  df-relexp 14914
This theorem is referenced by:  rtrclreclem2  14953  rtrclreclem4  14955
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