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Theorem relcoi1 6270
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 6256 . 2 (𝑅 ∘ ( I ↾ 𝑅)) = (𝑅 𝑅)
2 relresfld 6268 . 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 1533   cuni 4902   I cid 5566  cres 5671  ccom 5673  Rel wrel 5674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681
This theorem is referenced by:  relexpsucl  14981  tsrdir  18566
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