Theorem dfcoss2 38458

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 8625). 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 38456	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
2		elecALTV 38309	. . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝑥))
3	2	el2v 3443	. . . . 5 ⊢ (𝑥 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝑥)
4		elecALTV 38309	. . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑦 ∈ V) → (𝑦 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝑦))
5	4	el2v 3443	. . . . 5 ⊢ (𝑦 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝑦)
6	3, 5	anbi12i 628	. . . 4 ⊢ ((𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅) ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦))
7	6	exbii 1849	. . 3 ⊢ (∃𝑢(𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅) ↔ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦))
8	7	opabbii 5156	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
9	1, 8	eqtr4i 2757	1 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅)}

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  Vcvv 3436   class class class wbr 5089  {copab 5151  [cec 8620   ≀ ccoss 38223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ec 8624  df-coss 38456

This theorem is referenced by:  coss0  38524