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Theorem relcoi1 5581
 Description: Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.) (Proof shortened by OpenAI, 3-Jul-2020.)
Assertion
Ref Expression
relcoi1 (Rel 𝑅 → (𝑅 ∘ ( I ↾ 𝑅)) = 𝑅)

Proof of Theorem relcoi1
StepHypRef Expression
1 coires1 5570 . 2 (𝑅 ∘ ( I ↾ 𝑅)) = (𝑅 𝑅)
2 relresfld 5579 . 2 (Rel 𝑅 → (𝑅 𝑅) = 𝑅)
31, 2syl5eq 2656 1 (Rel 𝑅 → (𝑅 ∘ ( I ↾ 𝑅)) = 𝑅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  ∪ cuni 4372   I cid 4948   ↾ cres 5040   ∘ ccom 5042  Rel wrel 5043 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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 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-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 This theorem is referenced by:  relexpsucl  13621  tsrdir  17061
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