Theorem rnghmresel

Description: An element of the non-unital ring homomorphisms restricted to a subset of non-unital rings is a non-unital ring homomorphisms. (Contributed by AV, 9-Mar-2020)

	Ref	Expression
Hypothesis	rnghmresel.h	|- ( ph -> H = ( RngHom |` ( B X. B ) ) )
Assertion	rnghmresel	|- ( ( ph /\ ( X e. B /\ Y e. B ) /\ F e. ( X H Y ) ) -> F e. ( X RngHom Y ) )

Proof

Step	Hyp	Ref	Expression
1		rnghmresel.h	|- ( ph -> H = ( RngHom |` ( B X. B ) ) )
2	1	adantr	|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> H = ( RngHom |` ( B X. B ) ) )
3	2	oveqd	|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X H Y ) = ( X ( RngHom |` ( B X. B ) ) Y ) )
4		ovres	|- ( ( X e. B /\ Y e. B ) -> ( X ( RngHom |` ( B X. B ) ) Y ) = ( X RngHom Y ) )
5	4	adantl	|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X ( RngHom |` ( B X. B ) ) Y ) = ( X RngHom Y ) )
6	3 5	eqtrd	|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X H Y ) = ( X RngHom Y ) )
7	6	eleq2d	|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( F e. ( X H Y ) <-> F e. ( X RngHom Y ) ) )
8	7	biimp3a	|- ( ( ph /\ ( X e. B /\ Y e. B ) /\ F e. ( X H Y ) ) -> F e. ( X RngHom Y ) )