Theorem rngoiso1o

Description: A ring isomorphism is a bijection. (Contributed by Jeff Madsen, 16-Jun-2011)

	Ref	Expression
Hypotheses	rngisoval.1	⊢ 𝐺 = ( 1st ‘ 𝑅 )
	rngisoval.2	⊢ 𝑋 = ran 𝐺
	rngisoval.3	⊢ 𝐽 = ( 1st ‘ 𝑆 )
	rngisoval.4	⊢ 𝑌 = ran 𝐽
Assertion	rngoiso1o	⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) → 𝐹 : 𝑋 –1-1-onto→ 𝑌 )

Proof

Step	Hyp	Ref	Expression
1		rngisoval.1	⊢ 𝐺 = ( 1st ‘ 𝑅 )
2		rngisoval.2	⊢ 𝑋 = ran 𝐺
3		rngisoval.3	⊢ 𝐽 = ( 1st ‘ 𝑆 )
4		rngisoval.4	⊢ 𝑌 = ran 𝐽
5	1 2 3 4	isrngoiso	⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ) → ( 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐹 : 𝑋 –1-1-onto→ 𝑌 ) ) )
6	5	simplbda	⊢ ( ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ) ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) → 𝐹 : 𝑋 –1-1-onto→ 𝑌 )
7	6	3impa	⊢ ( ( 𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ) → 𝐹 : 𝑋 –1-1-onto→ 𝑌 )