Theorem dff12

Description: Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996)

	Ref	Expression
Assertion	dff12	$$⊢F:A\underset{1-1}{⟶}B\leftrightarrow \left(F:A⟶B\wedge \forall y{\exists }^{*}xxFy\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-f1	$$⊢F:A\underset{1-1}{⟶}B\leftrightarrow \left(F:A⟶B\wedge \mathrm{Fun}{F}^{-1}\right)$$
2		funcnv2	$$⊢\mathrm{Fun}{F}^{-1}\leftrightarrow \forall y{\exists }^{*}xxFy$$
3	2	anbi2i	$$⊢\left(F:A⟶B\wedge \mathrm{Fun}{F}^{-1}\right)\leftrightarrow \left(F:A⟶B\wedge \forall y{\exists }^{*}xxFy\right)$$
4	1 3	bitri	$$⊢F:A\underset{1-1}{⟶}B\leftrightarrow \left(F:A⟶B\wedge \forall y{\exists }^{*}xxFy\right)$$