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 C_ 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 C_ B ) -> dom F = A )
4		simpr	|- ( ( F Fn A /\ A C_ B ) -> A C_ B )
5	3 4	eqsstrd	|- ( ( F Fn A /\ A C_ B ) -> dom F C_ B )
6		relssres	|- ( ( Rel F /\ dom F C_ B ) -> ( F |` B ) = F )
7	1 5 6	syl2an2r	|- ( ( F Fn A /\ A C_ B ) -> ( F |` B ) = F )