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Theorem relexp0 15010
Description: A relation composed zero times is the (restricted) identity. (Contributed by RP, 22-May-2020.)
Assertion
Ref Expression
relexp0 ((𝑅𝑉 ∧ Rel 𝑅) → (𝑅𝑟0) = ( I ↾ 𝑅))

Proof of Theorem relexp0
StepHypRef Expression
1 relexp0g 15009 . 2 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
2 relfld 6284 . . . 4 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
32reseq2d 5989 . . 3 (Rel 𝑅 → ( I ↾ 𝑅) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
43eqcomd 2734 . 2 (Rel 𝑅 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ 𝑅))
51, 4sylan9eq 2788 1 ((𝑅𝑉 ∧ Rel 𝑅) → (𝑅𝑟0) = ( I ↾ 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  cun 3947   cuni 4912   I cid 5579  dom cdm 5682  ran crn 5683  cres 5684  Rel wrel 5687  (class class class)co 7426  0cc0 11146  𝑟crelexp 15006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-mulcl 11208  ax-i2m1 11214
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-n0 12511  df-relexp 15007
This theorem is referenced by:  relexp0d  15011  relexpsucl  15018  relexpsucr  15019  relexpindlem  15050
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