Theorem dmecd 38285

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, 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 2997	. . 3 ⊢ (𝜑 → ([𝐵]𝑅 ≠ ∅ ↔ [𝐶]𝑅 ≠ ∅))
3		ecdmn0 8792	. . 3 ⊢ (𝐵 ∈ dom 𝑅 ↔ [𝐵]𝑅 ≠ ∅)
4		ecdmn0 8792	. . 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 2824	. 2 ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐵 ∈ 𝐴))
8	6	eleq2d 2824	. 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 1536   ∈ wcel 2105   ≠ wne 2937  ∅c0 4338  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:  dmec2d  38286