Theorem f1resf1

Description: The restriction of an injective function is injective. (Contributed by AV, 28-Jun-2022)

	Ref	Expression
Assertion	f1resf1	$$⊢\left(F:A\underset{1-1}{⟶}B\wedge C\subseteq A\wedge \left({F\upharpoonright }_C\right):C⟶D\right)\to \left({F\upharpoonright }_C\right):C\underset{1-1}{⟶}D$$

Proof

Step	Hyp	Ref	Expression
1		f1ssres	$$⊢\left(F:A\underset{1-1}{⟶}B\wedge C\subseteq A\right)\to \left({F\upharpoonright }_C\right):C\underset{1-1}{⟶}B$$
2	1	3adant3	$$⊢\left(F:A\underset{1-1}{⟶}B\wedge C\subseteq A\wedge \left({F\upharpoonright }_C\right):C⟶D\right)\to \left({F\upharpoonright }_C\right):C\underset{1-1}{⟶}B$$
3		frn	$$⊢\left({F\upharpoonright }_C\right):C⟶D\to \mathrm{ran}\left({F\upharpoonright }_C\right)\subseteq D$$
4	3	3ad2ant3	$$⊢\left(F:A\underset{1-1}{⟶}B\wedge C\subseteq A\wedge \left({F\upharpoonright }_C\right):C⟶D\right)\to \mathrm{ran}\left({F\upharpoonright }_C\right)\subseteq D$$
5		f1ssr	$$⊢\left(\left({F\upharpoonright }_C\right):C\underset{1-1}{⟶}B\wedge \mathrm{ran}\left({F\upharpoonright }_C\right)\subseteq D\right)\to \left({F\upharpoonright }_C\right):C\underset{1-1}{⟶}D$$
6	2 4 5	syl2anc	$$⊢\left(F:A\underset{1-1}{⟶}B\wedge C\subseteq A\wedge \left({F\upharpoonright }_C\right):C⟶D\right)\to \left({F\upharpoonright }_C\right):C\underset{1-1}{⟶}D$$