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Theorem relcoels 38425
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 38424 . 2 Rel ≀ ( E ↾ 𝐴)
2 df-coels 38413 . . 3 𝐴 = ≀ ( E ↾ 𝐴)
32releqi 5787 . 2 (Rel ∼ 𝐴 ↔ Rel ≀ ( E ↾ 𝐴))
41, 3mpbir 231 1 Rel ∼ 𝐴
Colors of variables: wff setvar class
Syntax hints:   E cep 5583  ccnv 5684  cres 5687  Rel wrel 5690  ccoss 38182  ccoels 38183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-ss 3968  df-opab 5206  df-xp 5691  df-rel 5692  df-coss 38412  df-coels 38413
This theorem is referenced by:  erimeq2  38679
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