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Theorem eldmcoss2 36967
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 36966 . 2 (𝐴𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
2 brcoss 36939 . . . 4 ((𝐴𝑉𝐴𝑉) → (𝐴𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐴)))
32anidms 568 . . 3 (𝐴𝑉 → (𝐴𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐴)))
4 pm4.24 565 . . . 4 (𝑢𝑅𝐴 ↔ (𝑢𝑅𝐴𝑢𝑅𝐴))
54exbii 1851 . . 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 205  wa 397  wex 1782  wcel 2107   class class class wbr 5106  dom cdm 5634  ccoss 36680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-coss 36919
This theorem is referenced by:  refrelcosslem  36970
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