Mathbox for Peter Mazsa < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  relcoels Structured version   Visualization version   GIF version

Theorem relcoels 35705
 Description: Coelements on 𝐴 is a relation. (Contributed by Peter Mazsa, 5-Oct-2021.)
Assertion
Ref Expression
relcoels Rel ∼ 𝐴

Proof of Theorem relcoels
StepHypRef Expression
1 relcoss 35704 . 2 Rel ≀ ( E ↾ 𝐴)
2 df-coels 35696 . . 3 𝐴 = ≀ ( E ↾ 𝐴)
32releqi 5628 . 2 (Rel ∼ 𝐴 ↔ Rel ≀ ( E ↾ 𝐴))
41, 3mpbir 233 1 Rel ∼ 𝐴
 Colors of variables: wff setvar class Syntax hints:   E cep 5440  ◡ccnv 5530   ↾ cres 5533  Rel wrel 5536   ≀ ccoss 35489   ∼ ccoels 35490 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-rab 3134  df-v 3475  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-opab 5105  df-xp 5537  df-rel 5538  df-coss 35695  df-coels 35696 This theorem is referenced by:  erim2  35947
 Copyright terms: Public domain W3C validator