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Theorem relcoi2 4791
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 4538 . . . 4 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
2 unss 3111 . . . . 5 ((dom 𝑅 𝑅 ran 𝑅 𝑅) ↔ (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅)
3 simpr 103 . . . . 5 ((dom 𝑅 𝑅 ran 𝑅 𝑅) → ran 𝑅 𝑅)
42, 3sylbir 125 . . . 4 ((dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅 → ran 𝑅 𝑅)
51, 4ax-mp 7 . . 3 ran 𝑅 𝑅
6 cores 4767 . . 3 (ran 𝑅 𝑅 → (( I ↾ 𝑅) ∘ 𝑅) = ( I ∘ 𝑅))
75, 6mp1i 10 . 2 (Rel 𝑅 → (( I ↾ 𝑅) ∘ 𝑅) = ( I ∘ 𝑅))
8 coi2 4780 . 2 (Rel 𝑅 → ( I ∘ 𝑅) = 𝑅)
97, 8eqtrd 2069 1 (Rel 𝑅 → (( I ↾ 𝑅) ∘ 𝑅) = 𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  cun 2909  wss 2911   cuni 3571   I cid 4016  dom cdm 4288  ran crn 4289  cres 4290  ccom 4292  Rel wrel 4293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300
This theorem is referenced by: (None)
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