Theorem dfcoss3 38396

Description: Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 38393). (Contributed by Peter Mazsa, 27-Dec-2018.)

Assertion

Ref	Expression
dfcoss3	⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅)

Proof of Theorem dfcoss3

Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brcnvg 5893	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥))
2	1	el2v 3485	. . . . 5 ⊢ (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥)
3	2	anbi1i 624	. . . 4 ⊢ ((𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦) ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦))
4	3	exbii 1845	. . 3 ⊢ (∃𝑢(𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦) ↔ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦))
5	4	opabbii 5215	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
6		df-co 5698	. 2 ⊢ (𝑅 ∘ ◡𝑅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦)}
7		df-coss 38393	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
8	5, 6, 7	3eqtr4ri 2774	1 ⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537  ∃wex 1776  Vcvv 3478   class class class wbr 5148  {copab 5210  ^◡ccnv 5688   ∘ ccom 5693   ≀ ccoss 38162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-cnv 5697  df-co 5698  df-coss 38393

This theorem is referenced by:  cossex  38401  dmcoss3  38435  funALTVfun  38680