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	⊢ ( 𝜑 → 𝐻 = ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) )
Assertion	rnghmresel	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ) → 𝐹 ∈ ( 𝑋 RngHomo 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		rnghmresel.h	⊢ ( 𝜑 → 𝐻 = ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) )
2	1	adantr	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐻 = ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) )
3	2	oveqd	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 𝐻 𝑌 ) = ( 𝑋 ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) 𝑌 ) )
4		ovres	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) 𝑌 ) = ( 𝑋 RngHomo 𝑌 ) )
5	4	adantl	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( RngHomo ↾ ( 𝐵 × 𝐵 ) ) 𝑌 ) = ( 𝑋 RngHomo 𝑌 ) )
6	3 5	eqtrd	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 𝐻 𝑌 ) = ( 𝑋 RngHomo 𝑌 ) )
7	6	eleq2d	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ↔ 𝐹 ∈ ( 𝑋 RngHomo 𝑌 ) ) )
8	7	biimp3a	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ) → 𝐹 ∈ ( 𝑋 RngHomo 𝑌 ) )