Theorem funres11

Description: The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998)

	Ref	Expression
Assertion	funres11	$$⊢\mathrm{Fun}{F}^{-1}\to \mathrm{Fun}{\left({F\upharpoonright }_A\right)}^{-1}$$

Proof

Step	Hyp	Ref	Expression
1		resss	$$⊢{F\upharpoonright }_A\subseteq F$$
2		cnvss	$$⊢{F\upharpoonright }_A\subseteq F\to {\left({F\upharpoonright }_A\right)}^{-1}\subseteq F^{-1}$$
3		funss	$$⊢{\left({F\upharpoonright }_A\right)}^{-1}\subseteq F^{-1}\to \left(\mathrm{Fun}{F}^{-1}\to \mathrm{Fun}{\left({F\upharpoonright }_A\right)}^{-1}\right)$$
4	1 2 3	mp2b	$$⊢\mathrm{Fun}{F}^{-1}\to \mathrm{Fun}{\left({F\upharpoonright }_A\right)}^{-1}$$