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Metamath Proof Explorer

Theorem rimf1o

Description: An isomorphism of rings is a bijection. (Contributed by AV, 22-Oct-2019)

Ref Expression
Hypotheses rhmf1o.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
rhmf1o.c ⊢ 𝐶 = ( Base ‘ 𝑆 )
Assertion rimf1o ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐶 )

Proof

Step Hyp Ref Expression
1 rhmf1o.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 rhmf1o.c ⊢ 𝐶 = ( Base ‘ 𝑆 )
3 1 2 isrim ⊢ ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) )
4 3 simprbi ⊢ ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐶 )