Theorem cosscnv 37077

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 37072	. 2 ⊢ ≀ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢◡𝑅𝑥 ∧ 𝑢◡𝑅𝑦)}
2		brcnvg 5870	. . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑥 ∈ V) → (𝑢◡𝑅𝑥 ↔ 𝑥𝑅𝑢))
3	2	el2v 3480	. . . . 5 ⊢ (𝑢◡𝑅𝑥 ↔ 𝑥𝑅𝑢)
4		brcnvg 5870	. . . . . 6 ⊢ ((𝑢 ∈ V ∧ 𝑦 ∈ V) → (𝑢◡𝑅𝑦 ↔ 𝑦𝑅𝑢))
5	4	el2v 3480	. . . . 5 ⊢ (𝑢◡𝑅𝑦 ↔ 𝑦𝑅𝑢)
6	3, 5	anbi12i 627	. . . 4 ⊢ ((𝑢◡𝑅𝑥 ∧ 𝑢◡𝑅𝑦) ↔ (𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢))
7	6	exbii 1850	. . 3 ⊢ (∃𝑢(𝑢◡𝑅𝑥 ∧ 𝑢◡𝑅𝑦) ↔ ∃𝑢(𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢))
8	7	opabbii 5207	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢◡𝑅𝑥 ∧ 𝑢◡𝑅𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢)}
9	1, 8	eqtri 2759	1 ⊢ ≀ ◡𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢 ∧ 𝑦𝑅𝑢)}

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wex 1781  Vcvv 3472   class class class wbr 5140  {copab 5202  ^◡ccnv 5667   ≀ ccoss 36834

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 2702  ax-sep 5291  ax-nul 5298  ax-pr 5419

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 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3474  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5141  df-opab 5203  df-cnv 5676  df-coss 37072

This theorem is referenced by: (None)