Theorem rhmresfn

Description: The class of unital ring homomorphisms restricted to subsets of unital rings is a function. (Contributed by AV, 10-Mar-2020)

	Ref	Expression
Hypotheses	rhmresfn.b	|- ( ph -> B = ( U i^i Ring ) )
	rhmresfn.h	|- ( ph -> H = ( RingHom |` ( B X. B ) ) )
Assertion	rhmresfn	|- ( ph -> H Fn ( B X. B ) )

Proof

Step	Hyp	Ref	Expression
1		rhmresfn.b	|- ( ph -> B = ( U i^i Ring ) )
2		rhmresfn.h	|- ( ph -> H = ( RingHom |` ( B X. B ) ) )
3		rhmfn	|- RingHom Fn ( Ring X. Ring )
4		inss2	|- ( U i^i Ring ) C_ Ring
5	1 4	eqsstrdi	|- ( ph -> B C_ Ring )
6		xpss12	|- ( ( B C_ Ring /\ B C_ Ring ) -> ( B X. B ) C_ ( Ring X. Ring ) )
7	5 5 6	syl2anc	|- ( ph -> ( B X. B ) C_ ( Ring X. Ring ) )
8		fnssres	|- ( ( RingHom Fn ( Ring X. Ring ) /\ ( B X. B ) C_ ( Ring X. Ring ) ) -> ( RingHom |` ( B X. B ) ) Fn ( B X. B ) )
9	3 7 8	sylancr	|- ( ph -> ( RingHom |` ( B X. B ) ) Fn ( B X. B ) )
10	2	fneq1d	|- ( ph -> ( H Fn ( B X. B ) <-> ( RingHom |` ( B X. B ) ) Fn ( B X. B ) ) )
11	9 10	mpbird	|- ( ph -> H Fn ( B X. B ) )