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Metamath Proof Explorer

Theorem isrngim2

Description: An isomorphism of non-unital rings is a bijective homomorphism. (Contributed by AV, 23-Feb-2020)

Ref Expression
Hypotheses rnghmf1o.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
rnghmf1o.c ⊢ 𝐶 = ( Base ‘ 𝑆 )
Assertion isrngim2 ( ( 𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ) → ( 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 rnghmf1o.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 rnghmf1o.c ⊢ 𝐶 = ( Base ‘ 𝑆 )
3 isrngim ⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ) → ( 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ ◡ 𝐹 ∈ ( 𝑆 RngHom 𝑅 ) ) ) )
4 1 2 rnghmf1o ⊢ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) → ( 𝐹 : 𝐵 –1-1-onto→ 𝐶 ↔ ◡ 𝐹 ∈ ( 𝑆 RngHom 𝑅 ) ) )
5 4 bicomd ⊢ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) → ( ◡ 𝐹 ∈ ( 𝑆 RngHom 𝑅 ) ↔ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) )
6 5 a1i ⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ) → ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) → ( ◡ 𝐹 ∈ ( 𝑆 RngHom 𝑅 ) ↔ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) ) )
7 6 pm5.32d ⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ) → ( ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ ◡ 𝐹 ∈ ( 𝑆 RngHom 𝑅 ) ) ↔ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) ) )
8 3 7 bitrd ⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ) → ( 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) ) )