Theorem relcoels 38380

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 38379	. 2 ⊢ Rel ≀ (◡ E ↾ 𝐴)
2		df-coels 38368	. . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
3	2	releqi 5801	. 2 ⊢ (Rel ∼ 𝐴 ↔ Rel ≀ (◡ E ↾ 𝐴))
4	1, 3	mpbir 231	1 ⊢ Rel ∼ 𝐴

Syntax hints:   E cep 5598  ^◡ccnv 5699   ↾ cres 5702  Rel wrel 5705   ≀ ccoss 38135   ∼ ccoels 38136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-ss 3993  df-opab 5229  df-xp 5706  df-rel 5707  df-coss 38367  df-coels 38368

This theorem is referenced by:  erimeq2  38634