Theorem rngimf1o

Description: An isomorphism of non-unital rings is a bijection. (Contributed by AV, 23-Feb-2020)

	Ref	Expression
Hypotheses	rnghmf1o.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	rnghmf1o.c	⊢ 𝐶 = ( Base ‘ 𝑆 )
Assertion	rngimf1o	⊢ ( 𝐹 ∈ ( 𝑅 RngIsom 𝑆 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		rnghmf1o.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		rnghmf1o.c	⊢ 𝐶 = ( Base ‘ 𝑆 )
3		rngimrcl	⊢ ( 𝐹 ∈ ( 𝑅 RngIsom 𝑆 ) → ( 𝑅 ∈ V ∧ 𝑆 ∈ V ) )
4	1 2	isrngim	⊢ ( ( 𝑅 ∈ V ∧ 𝑆 ∈ V ) → ( 𝐹 ∈ ( 𝑅 RngIsom 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ∧ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) ) )
5		simpr	⊢ ( ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ∧ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐶 )
6	4 5	syl6bi	⊢ ( ( 𝑅 ∈ V ∧ 𝑆 ∈ V ) → ( 𝐹 ∈ ( 𝑅 RngIsom 𝑆 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) )
7	3 6	mpcom	⊢ ( 𝐹 ∈ ( 𝑅 RngIsom 𝑆 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐶 )