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Metamath Proof Explorer

Theorem gimf1o

Description: An isomorphism of groups is a bijection. (Contributed by Stefan O'Rear, 21-Jan-2015) (Revised by Mario Carneiro, 6-May-2015)

Ref Expression
Hypotheses isgim.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
isgim.c ⊢ 𝐶 = ( Base ‘ 𝑆 )
Assertion gimf1o ( 𝐹 ∈ ( 𝑅 GrpIso 𝑆 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐶 )

Proof

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