Theorem relcoels 35701

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 35700	. 2 ⊢ Rel ≀ (◡ E ↾ 𝐴)
2		df-coels 35692	. . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
3	2	releqi 5638	. 2 ⊢ (Rel ∼ 𝐴 ↔ Rel ≀ (◡ E ↾ 𝐴))
4	1, 3	mpbir 233	1 ⊢ Rel ∼ 𝐴

Syntax hints:   E cep 5450  ^◡ccnv 5540   ↾ cres 5543  Rel wrel 5546   ≀ ccoss 35485   ∼ ccoels 35486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3488  df-dif 3927  df-un 3929  df-in 3931  df-ss 3940  df-nul 4280  df-if 4454  df-sn 4554  df-pr 4556  df-op 4560  df-opab 5115  df-xp 5547  df-rel 5548  df-coss 35691  df-coels 35692

This theorem is referenced by:  erim2  35943