Theorem fnresdm

Description: A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004)

	Ref	Expression
Assertion	fnresdm	⊢ ( 𝐹 Fn 𝐴 → ( 𝐹 ↾ 𝐴 ) = 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		fnrel	⊢ ( 𝐹 Fn 𝐴 → Rel 𝐹 )
2		fndm	⊢ ( 𝐹 Fn 𝐴 → dom 𝐹 = 𝐴 )
3		eqimss	⊢ ( dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴 )
4	2 3	syl	⊢ ( 𝐹 Fn 𝐴 → dom 𝐹 ⊆ 𝐴 )
5		relssres	⊢ ( ( Rel 𝐹 ∧ dom 𝐹 ⊆ 𝐴 ) → ( 𝐹 ↾ 𝐴 ) = 𝐹 )
6	1 4 5	syl2anc	⊢ ( 𝐹 Fn 𝐴 → ( 𝐹 ↾ 𝐴 ) = 𝐹 )