Theorem isrngim

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

	Ref	Expression
Hypotheses	rnghmf1o.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	rnghmf1o.c	⊢ 𝐶 = ( Base ‘ 𝑆 )
Assertion	isrngim	⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ) → ( 𝐹 ∈ ( 𝑅 RngIsom 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 RngHomo 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		isrngisom	⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ) → ( 𝐹 ∈ ( 𝑅 RngIsom 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ∧ ◡ 𝐹 ∈ ( 𝑆 RngHomo 𝑅 ) ) ) )
4	1 2	rnghmf1o	⊢ ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) → ( 𝐹 : 𝐵 –1-1-onto→ 𝐶 ↔ ◡ 𝐹 ∈ ( 𝑆 RngHomo 𝑅 ) ) )
5	4	bicomd	⊢ ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) → ( ◡ 𝐹 ∈ ( 𝑆 RngHomo 𝑅 ) ↔ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) )
6	5	a1i	⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ) → ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) → ( ◡ 𝐹 ∈ ( 𝑆 RngHomo 𝑅 ) ↔ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) ) )
7	6	pm5.32d	⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ) → ( ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ∧ ◡ 𝐹 ∈ ( 𝑆 RngHomo 𝑅 ) ) ↔ ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ∧ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) ) )
8	3 7	bitrd	⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊 ) → ( 𝐹 ∈ ( 𝑅 RngIsom 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 RngHomo 𝑆 ) ∧ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) ) )