Theorem rnghmresfn

Description: The class of non-unital ring homomorphisms restricted to subsets of non-unital rings is a function. (Contributed by AV, 4-Mar-2020)

	Ref	Expression
Hypotheses	rnghmresfn.b	|- ( ph -> B = ( U i^i Rng ) )
	rnghmresfn.h	|- ( ph -> H = ( RngHomo |` ( B X. B ) ) )
Assertion	rnghmresfn	|- ( ph -> H Fn ( B X. B ) )

Proof

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