Theorem dmecd 38503

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 8688). (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 2991	. . 3 ⊢ (𝜑 → ([𝐵]𝑅 ≠ ∅ ↔ [𝐶]𝑅 ≠ ∅))
3		ecdmn0 8687	. . 3 ⊢ (𝐵 ∈ dom 𝑅 ↔ [𝐵]𝑅 ≠ ∅)
4		ecdmn0 8687	. . 3 ⊢ (𝐶 ∈ dom 𝑅 ↔ [𝐶]𝑅 ≠ ∅)
5	2, 3, 4	3bitr4g 314	. 2 ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐶 ∈ dom 𝑅))
6		dmecd.1	. . 3 ⊢ (𝜑 → dom 𝑅 = 𝐴)
7	6	eleq2d 2822	. 2 ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐵 ∈ 𝐴))
8	6	eleq2d 2822	. 2 ⊢ (𝜑 → (𝐶 ∈ dom 𝑅 ↔ 𝐶 ∈ 𝐴))
9	5, 7, 8	3bitr3d 309	1 ⊢ (𝜑 → (𝐵 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∅c0 4285  dom cdm 5624  [cec 8633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ec 8637

This theorem is referenced by:  dmec2d  38504