Theorem dmec2d 38774

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 8727). (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 2762	. 2 ⊢ (𝜑 → dom 𝑅 = dom 𝑅)
2		dmec2d.1	. 2 ⊢ (𝜑 → [𝐵]𝑅 = [𝐶]𝑅)
3	1, 2	dmecd 38773	1 ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐶 ∈ dom 𝑅))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1559   ∈ wcel 2141  dom cdm 5645  [cec 8671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5651  df-cnv 5653  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-ec 8675

This theorem is referenced by:  eqvrelth  39158