Theorem relcoels 37807

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 37806	. 2 ⊢ Rel ≀ (◡ E ↾ 𝐴)
2		df-coels 37795	. . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
3	2	releqi 5770	. 2 ⊢ (Rel ∼ 𝐴 ↔ Rel ≀ (◡ E ↾ 𝐴))
4	1, 3	mpbir 230	1 ⊢ Rel ∼ 𝐴

Syntax hints:   E cep 5572  ^◡ccnv 5668   ↾ cres 5671  Rel wrel 5674   ≀ ccoss 37556   ∼ ccoels 37557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-in 3950  df-ss 3960  df-opab 5204  df-xp 5675  df-rel 5676  df-coss 37794  df-coels 37795

This theorem is referenced by:  erimeq2  38061