Theorem dfcoss2 38377

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 8651). 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 38375	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
2		elecALTV 38228	. . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝑥))
3	2	el2v 3451	. . . . 5 ⊢ (𝑥 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝑥)
4		elecALTV 38228	. . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑦 ∈ V) → (𝑦 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝑦))
5	4	el2v 3451	. . . . 5 ⊢ (𝑦 ∈ [𝑢]𝑅 ↔ 𝑢𝑅𝑦)
6	3, 5	anbi12i 628	. . . 4 ⊢ ((𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅) ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦))
7	6	exbii 1848	. . 3 ⊢ (∃𝑢(𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅) ↔ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦))
8	7	opabbii 5169	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
9	1, 8	eqtr4i 2755	1 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥 ∈ [𝑢]𝑅 ∧ 𝑦 ∈ [𝑢]𝑅)}

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3444   class class class wbr 5102  {copab 5164  [cec 8646   ≀ ccoss 38142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ec 8650  df-coss 38375

This theorem is referenced by:  coss0  38443