Theorem cosscnv 38815

Description: Class of cosets by the converse of 𝑅. (Contributed by Peter Mazsa, 17-Jun-2020.)

Assertion

Ref	Expression
cosscnv	⊢ ≀ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢)}

Distinct variable group:   𝑢,𝑅,𝑥,𝑦

Proof of Theorem cosscnv

Step	Hyp	Ref	Expression
1		df-coss 38810	. 2 ⊢ ≀ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢◡𝑅𝑥 ∧ 𝑢◡𝑅𝑦)}
2		brcnvg 5823	. . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V) → (𝑢◡𝑅𝑥 ↔ 𝑥𝑅𝑢))
3	2	el2v 3434	. . . . 5 ⊢ (𝑢◡𝑅𝑥 ↔ 𝑥𝑅𝑢)
4		brcnvg 5823	. . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑦 ∈ V) → (𝑢◡𝑅𝑦 ↔ 𝑦𝑅𝑢))
5	4	el2v 3434	. . . . 5 ⊢ (𝑢◡𝑅𝑦 ↔ 𝑦𝑅𝑢)
6	3, 5	anbi12i 629	. . . 4 ⊢ ((𝑢◡𝑅𝑥 ∧ 𝑢◡𝑅𝑦) ↔ (𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢))
7	6	exbii 1850	. . 3 ⊢ (∃𝑢(𝑢◡𝑅𝑥 ∧ 𝑢◡𝑅𝑦) ↔ ∃𝑢(𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢))
8	7	opabbii 5141	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢◡𝑅𝑥 ∧ 𝑢◡𝑅𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢)}
9	1, 8	eqtri 2758	1 ⊢ ≀ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢)}

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781  Vcvv 3427   class class class wbr 5074  {copab 5136  ^◡ccnv 5619   ≀ ccoss 38492

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 2707  ax-sep 5220  ax-pr 5364

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 2714  df-cleq 2727  df-clel 2810  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-br 5075  df-opab 5137  df-cnv 5628  df-coss 38810

This theorem is referenced by: (None)