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Theorem eldmcoss2 35092
 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 35091 . 2 (𝐴𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴))
2 brcoss 35069 . . . 4 ((𝐴𝑉𝐴𝑉) → (𝐴𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐴)))
32anidms 559 . . 3 (𝐴𝑉 → (𝐴𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐴)))
4 pm4.24 556 . . . 4 (𝑢𝑅𝐴 ↔ (𝑢𝑅𝐴𝑢𝑅𝐴))
54exbii 1810 . . 3 (∃𝑢 𝑢𝑅𝐴 ↔ ∃𝑢(𝑢𝑅𝐴𝑢𝑅𝐴))
63, 5syl6bbr 281 . 2 (𝐴𝑉 → (𝐴𝑅𝐴 ↔ ∃𝑢 𝑢𝑅𝐴))
71, 6bitr4d 274 1 (𝐴𝑉 → (𝐴 ∈ dom ≀ 𝑅𝐴𝑅𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387  ∃wex 1742   ∈ wcel 2048   class class class wbr 4923  dom cdm 5400   ≀ ccoss 34845 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pr 5180 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-rab 3091  df-v 3411  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4924  df-opab 4986  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-coss 35052 This theorem is referenced by:  refrelcosslem  35095
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