Theorem rimf1o

Description: An isomorphism of rings is a bijection. (Contributed by Jeff Madsen, 16-Jun-2011) (Revised by AV, 22-Oct-2019)

	Ref	Expression
Hypotheses	rhmf1o.b	$$⊢B={\mathrm{Base}}_R$$
	rhmf1o.c	$$⊢C={\mathrm{Base}}_S$$
Assertion	rimf1o	$$⊢F\in \left(R\mathrm{RingIso}S\right)\to F:B\underset{1-1\; onto}{⟶}C$$

Proof

Step	Hyp	Ref	Expression
1		rhmf1o.b	$$⊢B={\mathrm{Base}}_R$$
2		rhmf1o.c	$$⊢C={\mathrm{Base}}_S$$
3	1 2	isrim	$$⊢F\in \left(R\mathrm{RingIso}S\right)\leftrightarrow \left(F\in \left(R\mathrm{RingHom}S\right)\wedge F:B\underset{1-1\; onto}{⟶}C\right)$$
4	3	simprbi	$$⊢F\in \left(R\mathrm{RingIso}S\right)\to F:B\underset{1-1\; onto}{⟶}C$$