Theorem dmec2d 38286

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 8793). (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 2735	. 2 ⊢ (𝜑 → dom 𝑅 = dom 𝑅)
2		dmec2d.1	. 2 ⊢ (𝜑 → [𝐵]𝑅 = [𝐶]𝑅)
3	1, 2	dmecd 38285	1 ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐶 ∈ dom 𝑅))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1536   ∈ wcel 2105  dom cdm 5688  [cec 8741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ec 8745

This theorem is referenced by:  eqvrelth  38592