Theorem cosscnv 38397

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 38392	. 2 ⊢ ≀ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢◡𝑅𝑥 ∧ 𝑢◡𝑅𝑦)}
2		brcnvg 5892	. . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V) → (𝑢◡𝑅𝑥 ↔ 𝑥𝑅𝑢))
3	2	el2v 3484	. . . . 5 ⊢ (𝑢◡𝑅𝑥 ↔ 𝑥𝑅𝑢)
4		brcnvg 5892	. . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑦 ∈ V) → (𝑢◡𝑅𝑦 ↔ 𝑦𝑅𝑢))
5	4	el2v 3484	. . . . 5 ⊢ (𝑢◡𝑅𝑦 ↔ 𝑦𝑅𝑢)
6	3, 5	anbi12i 628	. . . 4 ⊢ ((𝑢◡𝑅𝑥 ∧ 𝑢◡𝑅𝑦) ↔ (𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢))
7	6	exbii 1844	. . 3 ⊢ (∃𝑢(𝑢◡𝑅𝑥 ∧ 𝑢◡𝑅𝑦) ↔ ∃𝑢(𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢))
8	7	opabbii 5214	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢◡𝑅𝑥 ∧ 𝑢◡𝑅𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢)}
9	1, 8	eqtri 2762	1 ⊢ ≀ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢)}

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1536  ∃wex 1775  Vcvv 3477   class class class wbr 5147  {copab 5209  ^◡ccnv 5687   ≀ ccoss 38161

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-cnv 5696  df-coss 38392

This theorem is referenced by: (None)