Theorem dfcoss3 38825

Description: Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 38822). (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 5834	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥))
2	1	el2v 3436	. . . . 5 ⊢ (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥)
3	2	anbi1i 625	. . . 4 ⊢ ((𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦) ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦))
4	3	exbii 1850	. . 3 ⊢ (∃𝑢(𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦) ↔ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦))
5	4	opabbii 5152	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
6		df-co 5640	. 2 ⊢ (𝑅 ∘ ◡𝑅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦)}
7		df-coss 38822	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
8	5, 6, 7	3eqtr4ri 2770	1 ⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781  Vcvv 3429   class class class wbr 5085  {copab 5147  ^◡ccnv 5630   ∘ ccom 5635   ≀ ccoss 38504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-cnv 5639  df-co 5640  df-coss 38822

This theorem is referenced by:  cossex  38830  dmcoss3  38864  funALTVfun  39104