Theorem coss1cnvres 39631

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 39625	. 2 ⊢ ≀ ◡(𝑅 ↾ 𝐴) = {⟨𝑢, 𝑣⟩ ∣ ∃𝑥(𝑥◡(𝑅 ↾ 𝐴)𝑢 ∧ 𝑥◡(𝑅 ↾ 𝐴)𝑣)}
2		br1cnvres 39398	. . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑥◡(𝑅 ↾ 𝐴)𝑢 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥)))
3	2	elv 3459	. . . . . . 7 ⊢ (𝑥◡(𝑅 ↾ 𝐴)𝑢 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥))
4		br1cnvres 39398	. . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑥◡(𝑅 ↾ 𝐴)𝑣 ↔ (𝑣 ∈ 𝐴 ∧ 𝑣𝑅𝑥)))
5	4	elv 3459	. . . . . . 7 ⊢ (𝑥◡(𝑅 ↾ 𝐴)𝑣 ↔ (𝑣 ∈ 𝐴 ∧ 𝑣𝑅𝑥))
6	3, 5	anbi12i 640	. . . . . 6 ⊢ ((𝑥◡(𝑅 ↾ 𝐴)𝑢 ∧ 𝑥◡(𝑅 ↾ 𝐴)𝑣) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣𝑅𝑥)))
7		an4 669	. . . . . 6 ⊢ (((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥)) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣𝑅𝑥)))
8	6, 7	bitr4i 281	. . . . 5 ⊢ ((𝑥◡(𝑅 ↾ 𝐴)𝑢 ∧ 𝑥◡(𝑅 ↾ 𝐴)𝑣) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥)))
9	8	exbii 1881	. . . 4 ⊢ (∃𝑥(𝑥◡(𝑅 ↾ 𝐴)𝑢 ∧ 𝑥◡(𝑅 ↾ 𝐴)𝑣) ↔ ∃𝑥((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥)))
10		19.42v 1988	. . . 4 ⊢ (∃𝑥((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ (𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥)) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥)))
11	9, 10	bitri 278	. . 3 ⊢ (∃𝑥(𝑥◡(𝑅 ↾ 𝐴)𝑢 ∧ 𝑥◡(𝑅 ↾ 𝐴)𝑣) ↔ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥)))
12	11	opabbii 5175	. 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 2786	1 ⊢ ≀ ◡(𝑅 ↾ 𝐴) = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) ∧ ∃𝑥(𝑢𝑅𝑥 ∧ 𝑣𝑅𝑥))}

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  Vcvv 3454   class class class wbr 5106  {copab 5170  ^◡ccnv 5654   ↾ cres 5657   ≀ ccoss 39307

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-res 5667  df-coss 39625

This theorem is used by:  coss2cnvepres  39632