Theorem fconstfv

Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 . (Contributed by NM, 27-Aug-2004) (Proof shortened by OpenAI, 25-Mar-2020)

	Ref	Expression
Assertion	fconstfv	⊢ ( 𝐹 : 𝐴 ⟶ { 𝐵 } ↔ ( 𝐹 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ffnfv	⊢ ( 𝐹 : 𝐴 ⟶ { 𝐵 } ↔ ( 𝐹 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ { 𝐵 } ) )
2		fvex	⊢ ( 𝐹 ‘ 𝑥 ) ∈ V
3	2	elsn	⊢ ( ( 𝐹 ‘ 𝑥 ) ∈ { 𝐵 } ↔ ( 𝐹 ‘ 𝑥 ) = 𝐵 )
4	3	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ { 𝐵 } ↔ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝐵 )
5	4	anbi2i	⊢ ( ( 𝐹 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ { 𝐵 } ) ↔ ( 𝐹 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝐵 ) )
6	1 5	bitri	⊢ ( 𝐹 : 𝐴 ⟶ { 𝐵 } ↔ ( 𝐹 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝐵 ) )