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Theorem dmecd 37770
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 8767). (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 → (𝐵𝐴𝐶𝐴))

Proof of Theorem dmecd
StepHypRef Expression
1 dmecd.2 . . . 4 (𝜑 → [𝐵]𝑅 = [𝐶]𝑅)
21neeq1d 2996 . . 3 (𝜑 → ([𝐵]𝑅 ≠ ∅ ↔ [𝐶]𝑅 ≠ ∅))
3 ecdmn0 8766 . . 3 (𝐵 ∈ dom 𝑅 ↔ [𝐵]𝑅 ≠ ∅)
4 ecdmn0 8766 . . 3 (𝐶 ∈ dom 𝑅 ↔ [𝐶]𝑅 ≠ ∅)
52, 3, 43bitr4g 314 . 2 (𝜑 → (𝐵 ∈ dom 𝑅𝐶 ∈ dom 𝑅))
6 dmecd.1 . . 3 (𝜑 → dom 𝑅 = 𝐴)
76eleq2d 2815 . 2 (𝜑 → (𝐵 ∈ dom 𝑅𝐵𝐴))
86eleq2d 2815 . 2 (𝜑 → (𝐶 ∈ dom 𝑅𝐶𝐴))
95, 7, 83bitr3d 309 1 (𝜑 → (𝐵𝐴𝐶𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1534  wcel 2099  wne 2936  c0 4318  dom cdm 5672  [cec 8716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ec 8720
This theorem is referenced by:  dmec2d  37771
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