Theorem relcoels 37598

Description: Coelements on 𝐴 is a relation. (Contributed by Peter Mazsa, 5-Oct-2021.)

Assertion

Ref	Expression
relcoels	⊢ Rel ∼ 𝐴

Proof of Theorem relcoels

Step	Hyp	Ref	Expression
1		relcoss 37597	. 2 ⊢ Rel ≀ (◡ E ↾ 𝐴)
2		df-coels 37586	. . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
3	2	releqi 5777	. 2 ⊢ (Rel ∼ 𝐴 ↔ Rel ≀ (◡ E ↾ 𝐴))
4	1, 3	mpbir 230	1 ⊢ Rel ∼ 𝐴

Syntax hints:   E cep 5579  ^◡ccnv 5675   ↾ cres 5678  Rel wrel 5681   ≀ ccoss 37347   ∼ ccoels 37348

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3955  df-ss 3965  df-opab 5211  df-xp 5682  df-rel 5683  df-coss 37585  df-coels 37586

This theorem is referenced by:  erimeq2  37852