Theorem fconstopab 2448

Description: Representation of a constant function using ordered pairs.

Assertion

Ref	Expression
fconstopab	⊢ (A × {B}) = {⟨x, y⟩∣(x ∈ A ∧ y = B)}

Distinct variable group(s):   x,y,A   x,B,y

Proof of Theorem fconstopab

Step	Hyp	Ref	Expression
1		df-xp 2424	. 2 ⊢ (A × {B}) = {⟨x, y⟩∣(x ∈ A ∧ y ∈ {B})}
2		elsn 1820	. . . 4 ⊢ (y ∈ {B} ↔ y = B)
3	2	anbi2i 367	. . 3 ⊢ ((x ∈ A ∧ y ∈ {B}) ↔ (x ∈ A ∧ y = B))
4	3	biopabi 2103	. 2 ⊢ {⟨x, y⟩∣(x ∈ A ∧ y ∈ {B})} = {⟨x, y⟩∣(x ∈ A ∧ y = B)}
5	1, 4	eqtr 1119	1 ⊢ (A × {B}) = {⟨x, y⟩∣(x ∈ A ∧ y = B)}

Syntax hints:   ∧ wa 196   = wceq 1091   ∈ wcel 1092  {csn 1808  {copab 2055   × cxp 2408

This theorem is referenced by:  occllem5 5184

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-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-sn 1811  df-opab 2098  df-xp 2424

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