Theorem f1ssres

Description: A function that is one-to-one is also one-to-one on any subclass of its domain. (Contributed by Mario Carneiro, 17-Jan-2015)

	Ref	Expression
Assertion	f1ssres	$$⊢\left(F:A\underset{1-1}{⟶}B\wedge C\subseteq A\right)\to \left({F\upharpoonright }_C\right):C\underset{1-1}{⟶}B$$

Proof

Step	Hyp	Ref	Expression
1		f1f	$$⊢F:A\underset{1-1}{⟶}B\to F:A⟶B$$
2		fssres	$$⊢\left(F:A⟶B\wedge C\subseteq A\right)\to \left({F\upharpoonright }_C\right):C⟶B$$
3	1 2	sylan	$$⊢\left(F:A\underset{1-1}{⟶}B\wedge C\subseteq A\right)\to \left({F\upharpoonright }_C\right):C⟶B$$
4		df-f1	$$⊢F:A\underset{1-1}{⟶}B\leftrightarrow \left(F:A⟶B\wedge \mathrm{Fun}{F}^{-1}\right)$$
5		funres11	$$⊢\mathrm{Fun}{F}^{-1}\to \mathrm{Fun}{\left({F\upharpoonright }_C\right)}^{-1}$$
6	4 5	simplbiim	$$⊢F:A\underset{1-1}{⟶}B\to \mathrm{Fun}{\left({F\upharpoonright }_C\right)}^{-1}$$
7	6	adantr	$$⊢\left(F:A\underset{1-1}{⟶}B\wedge C\subseteq A\right)\to \mathrm{Fun}{\left({F\upharpoonright }_C\right)}^{-1}$$
8		df-f1	$$⊢\left({F\upharpoonright }_C\right):C\underset{1-1}{⟶}B\leftrightarrow \left(\left({F\upharpoonright }_C\right):C⟶B\wedge \mathrm{Fun}{\left({F\upharpoonright }_C\right)}^{-1}\right)$$
9	3 7 8	sylanbrc	$$⊢\left(F:A\underset{1-1}{⟶}B\wedge C\subseteq A\right)\to \left({F\upharpoonright }_C\right):C\underset{1-1}{⟶}B$$