Theorem cosscnv 39514

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 39509	. 2 ⊢ ≀ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢◡𝑅𝑥 ∧ 𝑢◡𝑅𝑦)}
2		brcnvg 5856	. . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V) → (𝑢◡𝑅𝑥 ↔ 𝑥𝑅𝑢))
3	2	el2v 3457	. . . . 5 ⊢ (𝑢◡𝑅𝑥 ↔ 𝑥𝑅𝑢)
4		brcnvg 5856	. . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑦 ∈ V) → (𝑢◡𝑅𝑦 ↔ 𝑦𝑅𝑢))
5	4	el2v 3457	. . . . 5 ⊢ (𝑢◡𝑅𝑦 ↔ 𝑦𝑅𝑢)
6	3, 5	anbi12i 640	. . . 4 ⊢ ((𝑢◡𝑅𝑥 ∧ 𝑢◡𝑅𝑦) ↔ (𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢))
7	6	exbii 1880	. . 3 ⊢ (∃𝑢(𝑢◡𝑅𝑥 ∧ 𝑢◡𝑅𝑦) ↔ ∃𝑢(𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢))
8	7	opabbii 5171	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢◡𝑅𝑥 ∧ 𝑢◡𝑅𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢)}
9	1, 8	eqtri 2783	1 ⊢ ≀ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢)}

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811  Vcvv 3450   class class class wbr 5102  {copab 5166  ^◡ccnv 5649   ≀ ccoss 39191

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-cnv 5658  df-coss 39509

This theorem is used by: (None)