Theorem funressnmo

Description: A function restricted to a singleton has at most one value for the singleton element as argument. (Contributed by AV, 2-Sep-2022)

	Ref	Expression
Assertion	funressnmo	⊢ ( ( 𝐴 ∈ 𝑉 ∧ Fun ( 𝐹 ↾ { 𝐴 } ) ) → ∃* 𝑦 𝐴 𝐹 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		sneq	⊢ ( 𝑥 = 𝐴 → { 𝑥 } = { 𝐴 } )
2	1	reseq2d	⊢ ( 𝑥 = 𝐴 → ( 𝐹 ↾ { 𝑥 } ) = ( 𝐹 ↾ { 𝐴 } ) )
3	2	funeqd	⊢ ( 𝑥 = 𝐴 → ( Fun ( 𝐹 ↾ { 𝑥 } ) ↔ Fun ( 𝐹 ↾ { 𝐴 } ) ) )
4		breq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝐹 𝑦 ↔ 𝐴 𝐹 𝑦 ) )
5	4	mobidv	⊢ ( 𝑥 = 𝐴 → ( ∃* 𝑦 𝑥 𝐹 𝑦 ↔ ∃* 𝑦 𝐴 𝐹 𝑦 ) )
6	3 5	imbi12d	⊢ ( 𝑥 = 𝐴 → ( ( Fun ( 𝐹 ↾ { 𝑥 } ) → ∃* 𝑦 𝑥 𝐹 𝑦 ) ↔ ( Fun ( 𝐹 ↾ { 𝐴 } ) → ∃* 𝑦 𝐴 𝐹 𝑦 ) ) )
7		funressnvmo	⊢ ( Fun ( 𝐹 ↾ { 𝑥 } ) → ∃* 𝑦 𝑥 𝐹 𝑦 )
8	6 7	vtoclg	⊢ ( 𝐴 ∈ 𝑉 → ( Fun ( 𝐹 ↾ { 𝐴 } ) → ∃* 𝑦 𝐴 𝐹 𝑦 ) )
9	8	imp	⊢ ( ( 𝐴 ∈ 𝑉 ∧ Fun ( 𝐹 ↾ { 𝐴 } ) ) → ∃* 𝑦 𝐴 𝐹 𝑦 )