ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  relcoi2 GIF version

Theorem relcoi2 5039
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 4772 . . . 4 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
2 unss 3220 . . . . 5 ((dom 𝑅 𝑅 ∧ ran 𝑅 𝑅) ↔ (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅)
3 simpr 109 . . . . 5 ((dom 𝑅 𝑅 ∧ ran 𝑅 𝑅) → ran 𝑅 𝑅)
42, 3sylbir 134 . . . 4 ((dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅 → ran 𝑅 𝑅)
51, 4ax-mp 5 . . 3 ran 𝑅 𝑅
6 cores 5012 . . 3 (ran 𝑅 𝑅 → (( I ↾ 𝑅) ∘ 𝑅) = ( I ∘ 𝑅))
75, 6mp1i 10 . 2 (Rel 𝑅 → (( I ↾ 𝑅) ∘ 𝑅) = ( I ∘ 𝑅))
8 coi2 5025 . 2 (Rel 𝑅 → ( I ∘ 𝑅) = 𝑅)
97, 8eqtrd 2150 1 (Rel 𝑅 → (( I ↾ 𝑅) ∘ 𝑅) = 𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  cun 3039  wss 3041   cuni 3706   I cid 4180  dom cdm 4509  ran crn 4510  cres 4511  ccom 4513  Rel wrel 4514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator