Theorem relcoels 38728

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 38727	. 2 ⊢ Rel ≀ (◡ E ↾ 𝐴)
2		df-coels 38716	. . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
3	2	releqi 5728	. 2 ⊢ (Rel ∼ 𝐴 ↔ Rel ≀ (◡ E ↾ 𝐴))
4	1, 3	mpbir 231	1 ⊢ Rel ∼ 𝐴

Syntax hints:   E cep 5524  ^◡ccnv 5624   ↾ cres 5627  Rel wrel 5630   ≀ ccoss 38397   ∼ ccoels 38398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3443  df-ss 3919  df-opab 5162  df-xp 5631  df-rel 5632  df-coss 38715  df-coels 38716

This theorem is referenced by:  erimeq2  38977