Theorem dmecd 38299

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, 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 2985	. . 3 ⊢ (𝜑 → ([𝐵]𝑅 ≠ ∅ ↔ [𝐶]𝑅 ≠ ∅))
3		ecdmn0 8726	. . 3 ⊢ (𝐵 ∈ dom 𝑅 ↔ [𝐵]𝑅 ≠ ∅)
4		ecdmn0 8726	. . 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 2815	. 2 ⊢ (𝜑 → (𝐵 ∈ dom 𝑅 ↔ 𝐵 ∈ 𝐴))
8	6	eleq2d 2815	. 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 1540   ∈ wcel 2109   ≠ wne 2926  ∅c0 4299  dom cdm 5641  [cec 8672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-xp 5647  df-cnv 5649  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ec 8676

This theorem is referenced by:  dmec2d  38300