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

Theorem relcoels 37936
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 37935 . 2 Rel ≀ ( E ↾ 𝐴)
2 df-coels 37924 . . 3 𝐴 = ≀ ( E ↾ 𝐴)
32releqi 5783 . 2 (Rel ∼ 𝐴 ↔ Rel ≀ ( E ↾ 𝐴))
41, 3mpbir 230 1 Rel ∼ 𝐴
Colors of variables: wff setvar class
Syntax hints:   E cep 5585  ccnv 5681  cres 5684  Rel wrel 5687  ccoss 37689  ccoels 37690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-in 3956  df-ss 3966  df-opab 5215  df-xp 5688  df-rel 5689  df-coss 37923  df-coels 37924
This theorem is referenced by:  erimeq2  38190
  Copyright terms: Public domain W3C validator