Theorem ec2 3203

Description: Alternate definition of R-coset of A. Definition 34 of [Suppes] p. 81.

Hypothesis

Ref	Expression
ec2.1	⊢ A ∈ V

Assertion

Ref	Expression
ec2	⊢ [A]R = {y∣ARy}

Distinct variable group(s):   y,A   y,R

Proof of Theorem ec2

Step	Hyp	Ref	Expression
1		dfima3 2605	. 2 ⊢ (R “ {A}) = {y∣∃x(x ∈ {A} ∧ ⟨x, y⟩ ∈ R)}
2		df-ec 3202	. 2 ⊢ [A]R = (R “ {A})
3		ec2.1	. . . . 5 ⊢ A ∈ V
4		opeq1 1876	. . . . . 6 ⊢ (x = A → ⟨x, y⟩ = ⟨A, y⟩)
5	4	eleq1d 1155	. . . . 5 ⊢ (x = A → (⟨x, y⟩ ∈ R ↔ ⟨A, y⟩ ∈ R))
6	3, 5	ceqsexv 1371	. . . 4 ⊢ (∃x(x = A ∧ ⟨x, y⟩ ∈ R) ↔ ⟨A, y⟩ ∈ R)
7		elsn 1820	. . . . . 6 ⊢ (x ∈ {A} ↔ x = A)
8	7	anbi1i 368	. . . . 5 ⊢ ((x ∈ {A} ∧ ⟨x, y⟩ ∈ R) ↔ (x = A ∧ ⟨x, y⟩ ∈ R))
9	8	biex 733	. . . 4 ⊢ (∃x(x ∈ {A} ∧ ⟨x, y⟩ ∈ R) ↔ ∃x(x = A ∧ ⟨x, y⟩ ∈ R))
10		df-br 2063	. . . 4 ⊢ (ARy ↔ ⟨A, y⟩ ∈ R)
11	6, 9, 10	3bitr4r 159	. . 3 ⊢ (ARy ↔ ∃x(x ∈ {A} ∧ ⟨x, y⟩ ∈ R))
12	11	biabi 1181	. 2 ⊢ {y∣ARy} = {y∣∃x(x ∈ {A} ∧ ⟨x, y⟩ ∈ R)}
13	1, 2, 12	3eqtr4 1126	1 ⊢ [A]R = {y∣ARy}

Syntax hints:   ∧ wa 196  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348  {csn 1808  ⟨cop 1810   class class class wbr 2054   “ cima 2413  [cec 3198

This theorem is referenced by:  elec 3216  ecid 3236

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-ec 3202

metamath.org