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Theorem relexp0 13559
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 13558 . 2 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
2 relfld 5563 . . . 4 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
32reseq2d 5303 . . 3 (Rel 𝑅 → ( I ↾ 𝑅) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
43eqcomd 2615 . 2 (Rel 𝑅 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ 𝑅))
51, 4sylan9eq 2663 1 ((𝑅𝑉 ∧ Rel 𝑅) → (𝑅𝑟0) = ( I ↾ 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  cun 3537   cuni 4366   I cid 4937  dom cdm 5027  ran crn 5028  cres 5029  Rel wrel 5032  (class class class)co 6526  0cc0 9792  𝑟crelexp 13556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-mulcl 9854  ax-i2m1 9860
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-iota 5753  df-fun 5791  df-fv 5797  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-n0 11142  df-relexp 13557
This theorem is referenced by:  relexp0d  13560  relexpsucr  13565  relexpsucl  13569  relexpindlem  13599
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