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Metamath Proof Explorer

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 )