Theorem fniniseg

Description: Membership in the preimage of a singleton, under a function. (Contributed by Mario Carneiro, 12-May-2014) (Proof shortened by Mario Carneiro , 28-Apr-2015)

	Ref	Expression
Assertion	fniniseg	⊢ ( 𝐹 Fn 𝐴 → ( 𝐶 ∈ ( ◡ 𝐹 “ { 𝐵 } ) ↔ ( 𝐶 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝐶 ) = 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		elpreima	⊢ ( 𝐹 Fn 𝐴 → ( 𝐶 ∈ ( ◡ 𝐹 “ { 𝐵 } ) ↔ ( 𝐶 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝐶 ) ∈ { 𝐵 } ) ) )
2		fvex	⊢ ( 𝐹 ‘ 𝐶 ) ∈ V
3	2	elsn	⊢ ( ( 𝐹 ‘ 𝐶 ) ∈ { 𝐵 } ↔ ( 𝐹 ‘ 𝐶 ) = 𝐵 )
4	3	anbi2i	⊢ ( ( 𝐶 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝐶 ) ∈ { 𝐵 } ) ↔ ( 𝐶 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝐶 ) = 𝐵 ) )
5	1 4	bitrdi	⊢ ( 𝐹 Fn 𝐴 → ( 𝐶 ∈ ( ◡ 𝐹 “ { 𝐵 } ) ↔ ( 𝐶 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝐶 ) = 𝐵 ) ) )