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Theorem relcoi1 6281
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 6267 . 2 (𝑅 ∘ ( I ↾ 𝑅)) = (𝑅 𝑅)
2 relresfld 6279 . 2 (Rel 𝑅 → (𝑅 𝑅) = 𝑅)
31, 2eqtrid 2778 1 (Rel 𝑅 → (𝑅 ∘ ( I ↾ 𝑅)) = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534   cuni 4907   I cid 5571  cres 5676  ccom 5678  Rel wrel 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3421  df-v 3466  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4325  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4908  df-br 5146  df-opab 5208  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686
This theorem is referenced by:  relexpsucl  15030  tsrdir  18623
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