Theorem dfcoss3 37272

Description: Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 37269). (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 5877	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥))
2	1	el2v 3482	. . . . 5 ⊢ (𝑥◡𝑅𝑢 ↔ 𝑢𝑅𝑥)
3	2	anbi1i 624	. . . 4 ⊢ ((𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦) ↔ (𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦))
4	3	exbii 1850	. . 3 ⊢ (∃𝑢(𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦) ↔ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦))
5	4	opabbii 5214	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
6		df-co 5684	. 2 ⊢ (𝑅 ∘ ◡𝑅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥◡𝑅𝑢 ∧ 𝑢𝑅𝑦)}
7		df-coss 37269	. 2 ⊢ ≀ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥 ∧ 𝑢𝑅𝑦)}
8	5, 6, 7	3eqtr4ri 2771	1 ⊢ ≀ 𝑅 = (𝑅 ∘ ◡𝑅)

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wex 1781  Vcvv 3474   class class class wbr 5147  {copab 5209  ^◡ccnv 5674   ∘ ccom 5679   ≀ ccoss 37031

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-cnv 5683  df-co 5684  df-coss 37269

This theorem is referenced by:  cossex  37277  dmcoss3  37311  funALTVfun  37556