Theorem dmec2d 38632

Description: Equality of the coset of 𝐵 and the coset of 𝐶 implies equivalence of domain elementhood (equivalence is not necessary as opposed to ereldm 8697). (Contributed by Peter Mazsa, 12-Oct-2018.)

Hypothesis

Ref	Expression
dmec2d.1	⊢ (𝜑 → [𝐵]𝑅 = [𝐶]𝑅)

Assertion

Ref	Expression
dmec2d	⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐶 ∈ dom 𝑅))

Proof of Theorem dmec2d

Step	Hyp	Ref	Expression
1		eqidd 2737	. 2 ⊢ (𝜑 → dom 𝑅 = dom 𝑅)
2		dmec2d.1	. 2 ⊢ (𝜑 → [𝐵]𝑅 = [𝐶]𝑅)
3	1, 2	dmecd 38631	1 ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐶 ∈ dom 𝑅))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  dom cdm 5631  [cec 8641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ec 8645

This theorem is referenced by:  eqvrelth  39016