Theorem relcoels 38894

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 38893	. 2 ⊢ Rel ≀ (◡ E ↾ 𝐴)
2		df-coels 38882	. . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
3	2	releqi 5723	. 2 ⊢ (Rel ∼ 𝐴 ↔ Rel ≀ (◡ E ↾ 𝐴))
4	1, 3	mpbir 233	1 ⊢ Rel ∼ 𝐴

Syntax hints:   E cep 5519  ^◡ccnv 5619   ↾ cres 5622  Rel wrel 5625   ≀ ccoss 38563   ∼ ccoels 38564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-v 3435  df-ss 3901  df-opab 5137  df-xp 5626  df-rel 5627  df-coss 38881  df-coels 38882

This theorem is referenced by:  erimeq2  39143