Theorem erimeq 39264

Description: Equivalence relation on its natural domain implies that the class of coelements on the domain is equal to the relation (this is the most convenient form of prter3 39507 and erimeq2 39263). (Contributed by Peter Mazsa, 7-Oct-2021.) (Revised by Peter Mazsa, 29-Dec-2021.)

Assertion

Ref	Expression
erimeq	⊢ (𝑅 ∈ 𝑉 → (𝑅 ErALTV 𝐴 → ∼ 𝐴 = 𝑅))

Proof of Theorem erimeq

Step	Hyp	Ref	Expression
1		dferALTV2 39253	. 2 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
2		erimeq2 39263	. 2 ⊢ (𝑅 ∈ 𝑉 → (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ∼ 𝐴 = 𝑅))
3	1, 2	biimtrid 244	1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ErALTV 𝐴 → ∼ 𝐴 = 𝑅))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1561   ∈ wcel 2143  dom cdm 5648   / cqs 8678   ∼ ccoels 38684   EqvRel weqvrel 38700   ErALTV werALTV 38709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-11 2192  ax-ext 2735  ax-sep 5247  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-id 5543  df-eprel 5548  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-ec 8681  df-qs 8685  df-coss 39001  df-coels 39002  df-refrel 39092  df-symrel 39124  df-trrel 39158  df-eqvrel 39169  df-dmqs 39223  df-erALTV 39249

This theorem is referenced by:  partimeq  39412