Theorem dfcoss2 36539

Description: Alternate definition of the class of cosets by 𝑅: 𝑥 and 𝑦 are cosets by 𝑅 iff there exists a set 𝑢 such that both 𝑥 and 𝑦 are are elements of the 𝑅-coset of 𝑢 (see also the comment of dfec2 8501). 𝑅 is usually a relation. (Contributed by Peter Mazsa, 16-Jan-2018.)

Assertion

Ref	Expression
dfcoss2	⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅)}

Distinct variable group:   𝑢,𝑅,𝑥,𝑦

Proof of Theorem dfcoss2

Step	Hyp	Ref	Expression
1		df-coss 36537	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
2		elecALTV 36405	. . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝑥))
3	2	el2v 3440	. . . . 5 ⊢ (𝑥 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝑥)
4		elecALTV 36405	. . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑦 ∈ V) → (𝑦 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝑦))
5	4	el2v 3440	. . . . 5 ⊢ (𝑦 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝑦)
6	3, 5	anbi12i 627	. . . 4 ⊢ ((𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅) ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦))
7	6	exbii 1850	. . 3 ⊢ (∃𝑢(𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅) ↔ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦))
8	7	opabbii 5141	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
9	1, 8	eqtr4i 2769	1 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅)}

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  Vcvv 3432   class class class wbr 5074  {copab 5136  [cec 8496   ≀ ccoss 36333

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ec 8500  df-coss 36537

This theorem is referenced by:  coss0  36597