Theorem fvresval

Description: The value of a function at a restriction is either null or the same as the function itself. (Contributed by Scott Fenton, 4-Sep-2011)

	Ref	Expression
Assertion	fvresval	⊢ ( ( ( 𝐹 ↾ 𝐵 ) ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) ∨ ( ( 𝐹 ↾ 𝐵 ) ‘ 𝐴 ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		exmid	⊢ ( 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 ∈ 𝐵 )
2		fvres	⊢ ( 𝐴 ∈ 𝐵 → ( ( 𝐹 ↾ 𝐵 ) ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )
3		nfvres	⊢ ( ¬ 𝐴 ∈ 𝐵 → ( ( 𝐹 ↾ 𝐵 ) ‘ 𝐴 ) = ∅ )
4	2 3	orim12i	⊢ ( ( 𝐴 ∈ 𝐵 ∨ ¬ 𝐴 ∈ 𝐵 ) → ( ( ( 𝐹 ↾ 𝐵 ) ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) ∨ ( ( 𝐹 ↾ 𝐵 ) ‘ 𝐴 ) = ∅ ) )
5	1 4	ax-mp	⊢ ( ( ( 𝐹 ↾ 𝐵 ) ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) ∨ ( ( 𝐹 ↾ 𝐵 ) ‘ 𝐴 ) = ∅ )