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 : 𝐵1-1-onto𝐶 )

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 : 𝐵1-1-onto𝐶 ) )
4 3 simprbi ( 𝐹 ∈ ( 𝑅 GrpIso 𝑆 ) → 𝐹 : 𝐵1-1-onto𝐶 )