Theorem coss1cnvres 38887

Description: Class of cosets by the converse of a restriction. (Contributed by Peter Mazsa, 8-Jun-2020.)

Assertion

Ref	Expression
coss1cnvres	⊢ ≀ ◡(𝑅 ↾ 𝐴) = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥))}

Distinct variable groups:   𝑢,𝐴,𝑣,𝑥   𝑢,𝑅,𝑣,𝑥

Proof of Theorem coss1cnvres

Step	Hyp	Ref	Expression
1		df-coss 38881	. 2 ⊢ ≀ ◡(𝑅 ↾ 𝐴) = {⟨𝑢, 𝑣⟩ ∣ ∃𝑥(𝑥◡(𝑅 ↾ 𝐴)𝑢 ∧ 𝑥◡(𝑅 ↾ 𝐴)𝑣)}
2		br1cnvres 38654	. . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑥◡(𝑅 ↾ 𝐴)𝑢 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥)))
3	2	elv 3438	. . . . . . 7 ⊢ (𝑥◡(𝑅 ↾ 𝐴)𝑢 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥))
4		br1cnvres 38654	. . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑥◡(𝑅 ↾ 𝐴)𝑣 ↔ (𝑣 ∈ 𝐴 ∧ 𝑣𝑅𝑥)))
5	4	elv 3438	. . . . . . 7 ⊢ (𝑥◡(𝑅 ↾ 𝐴)𝑣 ↔ (𝑣 ∈ 𝐴 ∧ 𝑣𝑅𝑥))
6	3, 5	anbi12i 635	. . . . . 6 ⊢ ((𝑥◡(𝑅 ↾ 𝐴)𝑢 ∧ 𝑥◡(𝑅 ↾ 𝐴)𝑣) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣𝑅𝑥)))
7		an4 663	. . . . . 6 ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥)) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣𝑅𝑥)))
8	6, 7	bitr4i 280	. . . . 5 ⊢ ((𝑥◡(𝑅 ↾ 𝐴)𝑢 ∧ 𝑥◡(𝑅 ↾ 𝐴)𝑣) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥)))
9	8	exbii 1856	. . . 4 ⊢ (∃𝑥(𝑥◡(𝑅 ↾ 𝐴)𝑢 ∧ 𝑥◡(𝑅 ↾ 𝐴)𝑣) ↔ ∃𝑥((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥)))
10		19.42v 1961	. . . 4 ⊢ (∃𝑥((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥)) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥)))
11	9, 10	bitri 277	. . 3 ⊢ (∃𝑥(𝑥◡(𝑅 ↾ 𝐴)𝑢 ∧ 𝑥◡(𝑅 ↾ 𝐴)𝑣) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥)))
12	11	opabbii 5141	. 2 ⊢ {⟨𝑢, 𝑣⟩ ∣ ∃𝑥(𝑥◡(𝑅 ↾ 𝐴)𝑢 ∧ 𝑥◡(𝑅 ↾ 𝐴)𝑣)} = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥))}
13	1, 12	eqtri 2764	1 ⊢ ≀ ◡(𝑅 ↾ 𝐴) = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥))}

Syntax hints:   ↔ wb 208   ∧ wa 397   = wceq 1548  ∃wex 1787   ∈ wcel 2121  Vcvv 3433   class class class wbr 5074  {copab 5136  ^◡ccnv 5619   ↾ cres 5622   ≀ ccoss 38563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-br 5075  df-opab 5137  df-xp 5626  df-rel 5627  df-cnv 5628  df-res 5632  df-coss 38881

This theorem is referenced by:  coss2cnvepres  38888