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Theorem relcoi2 6286
Description: Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.)
Assertion
Ref Expression
relcoi2 (Rel 𝑅 → (( I ↾ 𝑅) ∘ 𝑅) = 𝑅)

Proof of Theorem relcoi2
StepHypRef Expression
1 dmrnssfld 5977 . . 3 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
2 unss 4186 . . . 4 ((dom 𝑅 𝑅 ∧ ran 𝑅 𝑅) ↔ (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅)
3 simpr 483 . . . 4 ((dom 𝑅 𝑅 ∧ ran 𝑅 𝑅) → ran 𝑅 𝑅)
42, 3sylbir 234 . . 3 ((dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅 → ran 𝑅 𝑅)
5 cores 6258 . . 3 (ran 𝑅 𝑅 → (( I ↾ 𝑅) ∘ 𝑅) = ( I ∘ 𝑅))
61, 4, 5mp2b 10 . 2 (( I ↾ 𝑅) ∘ 𝑅) = ( I ∘ 𝑅)
7 coi2 6272 . 2 (Rel 𝑅 → ( I ∘ 𝑅) = 𝑅)
86, 7eqtrid 2780 1 (Rel 𝑅 → (( I ↾ 𝑅) ∘ 𝑅) = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  cun 3947  wss 3949   cuni 4912   I cid 5579  dom cdm 5682  ran crn 5683  cres 5684  ccom 5686  Rel wrel 5687
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-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694
This theorem is referenced by:  relexpsucr  15019  tsrdir  18603
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