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Theorem relexp0 13611
 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 13610 . 2 (𝑅𝑉 → (𝑅𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
2 relfld 5578 . . . 4 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
32reseq2d 5317 . . 3 (Rel 𝑅 → ( I ↾ 𝑅) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
43eqcomd 2616 . 2 (Rel 𝑅 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ 𝑅))
51, 4sylan9eq 2664 1 ((𝑅𝑉 ∧ Rel 𝑅) → (𝑅𝑟0) = ( I ↾ 𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∪ cun 3538  ∪ cuni 4372   I cid 4948  dom cdm 5038  ran crn 5039   ↾ cres 5040  Rel wrel 5043  (class class class)co 6549  0cc0 9815  ↑𝑟crelexp 13608 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-mulcl 9877  ax-i2m1 9883 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-n0 11170  df-relexp 13609 This theorem is referenced by:  relexp0d  13612  relexpsucr  13617  relexpsucl  13621  relexpindlem  13651
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