Theorem dmecd 37168

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 8750). (Contributed by Peter Mazsa, 9-Oct-2018.)

Hypotheses

Ref	Expression
dmecd.1	⊢ (𝜑 → dom 𝑅 = 𝐴)
dmecd.2	⊢ (𝜑 → [𝐵]𝑅 = [𝐶]𝑅)

Assertion

Ref	Expression
dmecd	⊢ (𝜑 → (𝐵 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))

Proof of Theorem dmecd

Step	Hyp	Ref	Expression
1		dmecd.2	. . . 4 ⊢ (𝜑 → [𝐵]𝑅 = [𝐶]𝑅)
2	1	neeq1d 3000	. . 3 ⊢ (𝜑 → ([𝐵]𝑅 ≠ ∅ ↔ [𝐶]𝑅 ≠ ∅))
3		ecdmn0 8749	. . 3 ⊢ (𝐵 ∈ dom 𝑅 ↔ [𝐵]𝑅 ≠ ∅)
4		ecdmn0 8749	. . 3 ⊢ (𝐶 ∈ dom 𝑅 ↔ [𝐶]𝑅 ≠ ∅)
5	2, 3, 4	3bitr4g 313	. 2 ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐶 ∈ dom 𝑅))
6		dmecd.1	. . 3 ⊢ (𝜑 → dom 𝑅 = 𝐴)
7	6	eleq2d 2819	. 2 ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐵 ∈ 𝐴))
8	6	eleq2d 2819	. 2 ⊢ (𝜑 → (𝐶 ∈ dom 𝑅 ↔ 𝐶 ∈ 𝐴))
9	5, 7, 8	3bitr3d 308	1 ⊢ (𝜑 → (𝐵 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∅c0 4322  dom cdm 5676  [cec 8700

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ec 8704

This theorem is referenced by:  dmec2d  37169