Theorem fnresdmss

Description: A function does not change when restricted to a set that contains its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Assertion	fnresdmss	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐹 ↾ 𝐵 ) = 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		fnrel	⊢ ( 𝐹 Fn 𝐴 → Rel 𝐹 )
2		fndm	⊢ ( 𝐹 Fn 𝐴 → dom 𝐹 = 𝐴 )
3	2	adantr	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐵 ) → dom 𝐹 = 𝐴 )
4		simpr	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 ⊆ 𝐵 )
5	3 4	eqsstrd	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐵 ) → dom 𝐹 ⊆ 𝐵 )
6		relssres	⊢ ( ( Rel 𝐹 ∧ dom 𝐹 ⊆ 𝐵 ) → ( 𝐹 ↾ 𝐵 ) = 𝐹 )
7	1 5 6	syl2an2r	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐹 ↾ 𝐵 ) = 𝐹 )