Theorem dmec2d 38685

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 8694). (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 2741	. 2 ⊢ (𝜑 → dom 𝑅 = dom 𝑅)
2		dmec2d.1	. 2 ⊢ (𝜑 → [𝐵]𝑅 = [𝐶]𝑅)
3	1, 2	dmecd 38684	1 ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐶 ∈ dom 𝑅))

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547   ∈ wcel 2119  dom cdm 5625  [cec 8638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-xp 5631  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ec 8642

This theorem is referenced by:  eqvrelth  39069