Theorem funssfv

Description: The value of a member of the domain of a subclass of a function. (Contributed by NM, 15-Aug-1994)

	Ref	Expression
Assertion	funssfv	⊢ ( ( Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ∈ dom 𝐺 ) → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		fvres	⊢ ( 𝐴 ∈ dom 𝐺 → ( ( 𝐹 ↾ dom 𝐺 ) ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )
2	1	eqcomd	⊢ ( 𝐴 ∈ dom 𝐺 → ( 𝐹 ‘ 𝐴 ) = ( ( 𝐹 ↾ dom 𝐺 ) ‘ 𝐴 ) )
3		funssres	⊢ ( ( Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ) → ( 𝐹 ↾ dom 𝐺 ) = 𝐺 )
4	3	fveq1d	⊢ ( ( Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ) → ( ( 𝐹 ↾ dom 𝐺 ) ‘ 𝐴 ) = ( 𝐺 ‘ 𝐴 ) )
5	2 4	sylan9eqr	⊢ ( ( ( Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ) ∧ 𝐴 ∈ dom 𝐺 ) → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝐴 ) )
6	5	3impa	⊢ ( ( Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ∈ dom 𝐺 ) → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝐴 ) )