Metamath Proof Explorer


Theorem isrim

Description: An isomorphism of rings is a bijective homomorphism. (Contributed by AV, 22-Oct-2019) Remove sethood antecedent. (Revised by SN, 12-Jan-2025)

Ref Expression
Hypotheses rhmf1o.b 𝐵 = ( Base ‘ 𝑅 )
rhmf1o.c 𝐶 = ( Base ‘ 𝑆 )
Assertion isrim ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐹 : 𝐵1-1-onto𝐶 ) )

Proof

Step Hyp Ref Expression
1 rhmf1o.b 𝐵 = ( Base ‘ 𝑅 )
2 rhmf1o.c 𝐶 = ( Base ‘ 𝑆 )
3 isrim0 ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐹 ∈ ( 𝑆 RingHom 𝑅 ) ) )
4 1 2 rhmf1o ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ( 𝐹 : 𝐵1-1-onto𝐶 𝐹 ∈ ( 𝑆 RingHom 𝑅 ) ) )
5 4 bicomd ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ( 𝐹 ∈ ( 𝑆 RingHom 𝑅 ) ↔ 𝐹 : 𝐵1-1-onto𝐶 ) )
6 5 pm5.32i ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐹 ∈ ( 𝑆 RingHom 𝑅 ) ) ↔ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐹 : 𝐵1-1-onto𝐶 ) )
7 3 6 bitri ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐹 : 𝐵1-1-onto𝐶 ) )