Theorem srngf1o

Description: The involution function in a star ring is a bijection. (Contributed by Mario Carneiro, 6-Oct-2015)

	Ref	Expression
Hypotheses	srngcnv.i	$$⊢\underset{\dot{}}{\ast }={\ast }_{\mathrm{𝑟𝑓}}\left(R\right)$$
	srngf1o.b	$$⊢B={\mathrm{Base}}_R$$
Assertion	srngf1o	$$⊢R\in \mathrm{*-Ring}\to \underset{\dot{}}{\ast }:B\underset{1-1\; onto}{⟶}B$$

Proof

Step	Hyp	Ref	Expression
1		srngcnv.i	$$⊢\underset{\dot{}}{\ast }={\ast }_{\mathrm{𝑟𝑓}}\left(R\right)$$
2		srngf1o.b	$$⊢B={\mathrm{Base}}_R$$
3		eqid	$$⊢{opp}_r\left(R\right)={opp}_r\left(R\right)$$
4	3 1	srngrhm	$$⊢R\in \mathrm{*-Ring}\to \underset{\dot{}}{\ast }\in \left(R\mathrm{RingHom}{opp}_r\left(R\right)\right)$$
5		eqid	$$⊢{\mathrm{Base}}_{{opp}_r\left(R\right)}={\mathrm{Base}}_{{opp}_r\left(R\right)}$$
6	2 5	rhmf	$$⊢\underset{\dot{}}{\ast }\in \left(R\mathrm{RingHom}{opp}_r\left(R\right)\right)\to \underset{\dot{}}{\ast }:B⟶{\mathrm{Base}}_{{opp}_r\left(R\right)}$$
7		ffn	$$⊢\underset{\dot{}}{\ast }:B⟶{\mathrm{Base}}_{{opp}_r\left(R\right)}\to \underset{\dot{}}{\ast }FnB$$
8	4 6 7	3syl	$$⊢R\in \mathrm{*-Ring}\to \underset{\dot{}}{\ast }FnB$$
9	1	srngcnv	$$⊢R\in \mathrm{*-Ring}\to \underset{\dot{}}{\ast }={\underset{\dot{}}{\ast }}^{-1}$$
10	9	fneq1d	$$⊢R\in \mathrm{*-Ring}\to \left(\underset{\dot{}}{\ast }FnB\leftrightarrow {\underset{\dot{}}{\ast }}^{-1}FnB\right)$$
11	8 10	mpbid	$$⊢R\in \mathrm{*-Ring}\to {\underset{\dot{}}{\ast }}^{-1}FnB$$
12		dff1o4	$$⊢\underset{\dot{}}{\ast }:B\underset{1-1\; onto}{⟶}B\leftrightarrow \left(\underset{\dot{}}{\ast }FnB\wedge {\underset{\dot{}}{\ast }}^{-1}FnB\right)$$
13	8 11 12	sylanbrc	$$⊢R\in \mathrm{*-Ring}\to \underset{\dot{}}{\ast }:B\underset{1-1\; onto}{⟶}B$$