Theorem dfcoss3 36277

Description: Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 36274). (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 5748	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥))
2	1	el2v 3416	. . . . 5 ⊢ (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥)
3	2	anbi1i 627	. . . 4 ⊢ ((𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦) ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦))
4	3	exbii 1855	. . 3 ⊢ (∃𝑢(𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦) ↔ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦))
5	4	opabbii 5120	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
6		df-co 5560	. 2 ⊢ (𝑅 ∘ ◡𝑅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦)}
7		df-coss 36274	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
8	5, 6, 7	3eqtr4ri 2776	1 ⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅)

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1543  ∃wex 1787  Vcvv 3408   class class class wbr 5053  {copab 5115  ^◡ccnv 5550   ∘ ccom 5555   ≀ ccoss 36070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-cnv 5559  df-co 5560  df-coss 36274

This theorem is referenced by:  cossex  36279  dmcoss3  36308  funALTVfun  36546