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 )