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Theorem eldmcoss2 38440
Description: Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 28-Dec-2018.)
Assertion
Ref Expression
eldmcoss2 (𝐴𝑉 → (𝐴 ∈ dom ≀ 𝑅𝐴𝑅𝐴))

Proof of Theorem eldmcoss2
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 eldmcoss 38439 . 2 (𝐴𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
2 brcoss 38412 . . . 4 ((𝐴𝑉𝐴𝑉) → (𝐴𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐴)))
32anidms 566 . . 3 (𝐴𝑉 → (𝐴𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐴)))
4 pm4.24 563 . . . 4 (𝑢𝑅𝐴 ↔ (𝑢𝑅𝐴𝑢𝑅𝐴))
54exbii 1844 . . 3 (∃𝑢 𝑢𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐴))
63, 5bitr4di 289 . 2 (𝐴𝑉 → (𝐴𝑅𝐴 ↔ ∃𝑢 𝑢𝑅𝐴))
71, 6bitr4d 282 1 (𝐴𝑉 → (𝐴 ∈ dom ≀ 𝑅𝐴𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wex 1775  wcel 2105   class class class wbr 5147  dom cdm 5688  ccoss 38161
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-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-rab 3433  df-v 3479  df-dif 3965  df-un 3967  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-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-coss 38392
This theorem is referenced by:  refrelcosslem  38443
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