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Theorem relcoi2 5882
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 5588 . . 3 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
2 unss 3985 . . . 4 ((dom 𝑅 𝑅 ∧ ran 𝑅 𝑅) ↔ (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅)
3 simpr 478 . . . 4 ((dom 𝑅 𝑅 ∧ ran 𝑅 𝑅) → ran 𝑅 𝑅)
42, 3sylbir 227 . . 3 ((dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅 → ran 𝑅 𝑅)
5 cores 5857 . . 3 (ran 𝑅 𝑅 → (( I ↾ 𝑅) ∘ 𝑅) = ( I ∘ 𝑅))
61, 4, 5mp2b 10 . 2 (( I ↾ 𝑅) ∘ 𝑅) = ( I ∘ 𝑅)
7 coi2 5871 . 2 (Rel 𝑅 → ( I ∘ 𝑅) = 𝑅)
86, 7syl5eq 2845 1 (Rel 𝑅 → (( I ↾ 𝑅) ∘ 𝑅) = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  cun 3767  wss 3769   cuni 4628   I cid 5219  dom cdm 5312  ran crn 5313  cres 5314  ccom 5316  Rel wrel 5317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324
This theorem is referenced by:  relexpsucr  14110  tsrdir  17553
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