Theorem relcoels 38421

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 38420	. 2 ⊢ Rel ≀ (◡ E ↾ 𝐴)
2		df-coels 38409	. . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
3	2	releqi 5721	. 2 ⊢ (Rel ∼ 𝐴 ↔ Rel ≀ (◡ E ↾ 𝐴))
4	1, 3	mpbir 231	1 ⊢ Rel ∼ 𝐴

Syntax hints:   E cep 5518  ^◡ccnv 5618   ↾ cres 5621  Rel wrel 5624   ≀ ccoss 38175   ∼ ccoels 38176

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3438  df-ss 3920  df-opab 5155  df-xp 5625  df-rel 5626  df-coss 38408  df-coels 38409

This theorem is referenced by:  erimeq2  38676