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Theorem relcoi1 6103
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 6091 . 2 (𝑅 ∘ ( I ↾ 𝑅)) = (𝑅 𝑅)
2 relresfld 6101 . 2 (Rel 𝑅 → (𝑅 𝑅) = 𝑅)
31, 2syl5eq 2867 1 (Rel 𝑅 → (𝑅 ∘ ( I ↾ 𝑅)) = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537   cuni 4812   I cid 5433  cres 5531  ccom 5533  Rel wrel 5534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5177  ax-nul 5184  ax-pr 5304
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-nul 4268  df-if 4442  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4813  df-br 5041  df-opab 5103  df-id 5434  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541
This theorem is referenced by:  relexpsucl  14370  tsrdir  17824
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