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Theorem relcoi2 6106
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 5819 . . 3 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
2 unss 4135 . . . 4 ((dom 𝑅 𝑅 ∧ ran 𝑅 𝑅) ↔ (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅)
3 simpr 488 . . . 4 ((dom 𝑅 𝑅 ∧ ran 𝑅 𝑅) → ran 𝑅 𝑅)
42, 3sylbir 238 . . 3 ((dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅 → ran 𝑅 𝑅)
5 cores 6080 . . 3 (ran 𝑅 𝑅 → (( I ↾ 𝑅) ∘ 𝑅) = ( I ∘ 𝑅))
61, 4, 5mp2b 10 . 2 (( I ↾ 𝑅) ∘ 𝑅) = ( I ∘ 𝑅)
7 coi2 6094 . 2 (Rel 𝑅 → ( I ∘ 𝑅) = 𝑅)
86, 7syl5eq 2869 1 (Rel 𝑅 → (( I ↾ 𝑅) ∘ 𝑅) = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  cun 3906  wss 3908   cuni 4813   I cid 5436  dom cdm 5532  ran crn 5533  cres 5534  ccom 5536  Rel wrel 5537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544
This theorem is referenced by:  relexpsucr  14379  tsrdir  17839
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