Theorem coss1cnvres 38393

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 38387	. 2 ⊢ ≀ ◡(𝑅 ↾ 𝐴) = {⟨𝑢, 𝑣⟩ ∣ ∃𝑥(𝑥◡(𝑅 ↾ 𝐴)𝑢 ∧ 𝑥◡(𝑅 ↾ 𝐴)𝑣)}
2		br1cnvres 38245	. . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑥◡(𝑅 ↾ 𝐴)𝑢 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥)))
3	2	elv 3468	. . . . . . 7 ⊢ (𝑥◡(𝑅 ↾ 𝐴)𝑢 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥))
4		br1cnvres 38245	. . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑥◡(𝑅 ↾ 𝐴)𝑣 ↔ (𝑣 ∈ 𝐴 ∧ 𝑣𝑅𝑥)))
5	4	elv 3468	. . . . . . 7 ⊢ (𝑥◡(𝑅 ↾ 𝐴)𝑣 ↔ (𝑣 ∈ 𝐴 ∧ 𝑣𝑅𝑥))
6	3, 5	anbi12i 628	. . . . . 6 ⊢ ((𝑥◡(𝑅 ↾ 𝐴)𝑢 ∧ 𝑥◡(𝑅 ↾ 𝐴)𝑣) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣𝑅𝑥)))
7		an4 656	. . . . . 6 ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥)) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣𝑅𝑥)))
8	6, 7	bitr4i 278	. . . . 5 ⊢ ((𝑥◡(𝑅 ↾ 𝐴)𝑢 ∧ 𝑥◡(𝑅 ↾ 𝐴)𝑣) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥)))
9	8	exbii 1847	. . . 4 ⊢ (∃𝑥(𝑥◡(𝑅 ↾ 𝐴)𝑢 ∧ 𝑥◡(𝑅 ↾ 𝐴)𝑣) ↔ ∃𝑥((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥)))
10		19.42v 1952	. . . 4 ⊢ (∃𝑥((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥)) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥)))
11	9, 10	bitri 275	. . 3 ⊢ (∃𝑥(𝑥◡(𝑅 ↾ 𝐴)𝑢 ∧ 𝑥◡(𝑅 ↾ 𝐴)𝑣) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥)))
12	11	opabbii 5190	. 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 2757	1 ⊢ ≀ ◡(𝑅 ↾ 𝐴) = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥))}

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107  Vcvv 3463   class class class wbr 5123  {copab 5185  ^◡ccnv 5664   ↾ cres 5667   ≀ ccoss 38157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5124  df-opab 5186  df-xp 5671  df-rel 5672  df-cnv 5673  df-res 5677  df-coss 38387

This theorem is referenced by:  coss2cnvepres  38394