Theorem rngisomfv1

Description: If there is a non-unital ring isomorphism between a unital ring and a non-unital ring, then the function value of the ring unity of the unital ring is an element of the base set of the non-unital ring. (Contributed by AV, 27-Feb-2025)

	Ref	Expression
Hypotheses	rngisom1.1	⊢ 1 = ( 1r ‘ 𝑅 )
	rngisom1.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
Assertion	rngisomfv1	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) → ( 𝐹 ‘ 1 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		rngisom1.1	⊢ 1 = ( 1r ‘ 𝑅 )
2		rngisom1.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
3		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
4	3 2	rngimf1o	⊢ ( 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) → 𝐹 : ( Base ‘ 𝑅 ) –1-1-onto→ 𝐵 )
5		f1of	⊢ ( 𝐹 : ( Base ‘ 𝑅 ) –1-1-onto→ 𝐵 → 𝐹 : ( Base ‘ 𝑅 ) ⟶ 𝐵 )
6	4 5	syl	⊢ ( 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) → 𝐹 : ( Base ‘ 𝑅 ) ⟶ 𝐵 )
7	6	adantl	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) → 𝐹 : ( Base ‘ 𝑅 ) ⟶ 𝐵 )
8	3 1	ringidcl	⊢ ( 𝑅 ∈ Ring → 1 ∈ ( Base ‘ 𝑅 ) )
9	8	adantr	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) → 1 ∈ ( Base ‘ 𝑅 ) )
10	7 9	ffvelcdmd	⊢ ( ( 𝑅 ∈ Ring ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) → ( 𝐹 ‘ 1 ) ∈ 𝐵 )