Theorem dff14a

Description: A one-to-one function in terms of different function values for different arguments. (Contributed by Alexander van der Vekens, 26-Jan-2018)

	Ref	Expression
Assertion	dff14a	$$⊢F:A\underset{1-1}{⟶}B\leftrightarrow \left(F:A⟶B\wedge \forall x\in A\forall y\in A\left(x\ne y\to F\left(x\right)\ne F\left(y\right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		dff13	$$⊢F:A\underset{1-1}{⟶}B\leftrightarrow \left(F:A⟶B\wedge \forall x\in A\forall y\in A\left(F\left(x\right)=F\left(y\right)\to x=y\right)\right)$$
2		con34b	$$⊢\left(F\left(x\right)=F\left(y\right)\to x=y\right)\leftrightarrow \left(\neg x=y\to \neg F\left(x\right)=F\left(y\right)\right)$$
3		df-ne	$$⊢x\ne y\leftrightarrow \neg x=y$$
4	3	bicomi	$$⊢\neg x=y\leftrightarrow x\ne y$$
5		df-ne	$$⊢F\left(x\right)\ne F\left(y\right)\leftrightarrow \neg F\left(x\right)=F\left(y\right)$$
6	5	bicomi	$$⊢\neg F\left(x\right)=F\left(y\right)\leftrightarrow F\left(x\right)\ne F\left(y\right)$$
7	4 6	imbi12i	$$⊢\left(\neg x=y\to \neg F\left(x\right)=F\left(y\right)\right)\leftrightarrow \left(x\ne y\to F\left(x\right)\ne F\left(y\right)\right)$$
8	2 7	bitri	$$⊢\left(F\left(x\right)=F\left(y\right)\to x=y\right)\leftrightarrow \left(x\ne y\to F\left(x\right)\ne F\left(y\right)\right)$$
9	8	2ralbii	$$⊢\forall x\in A\forall y\in A\left(F\left(x\right)=F\left(y\right)\to x=y\right)\leftrightarrow \forall x\in A\forall y\in A\left(x\ne y\to F\left(x\right)\ne F\left(y\right)\right)$$
10	9	anbi2i	$$⊢\left(F:A⟶B\wedge \forall x\in A\forall y\in A\left(F\left(x\right)=F\left(y\right)\to x=y\right)\right)\leftrightarrow \left(F:A⟶B\wedge \forall x\in A\forall y\in A\left(x\ne y\to F\left(x\right)\ne F\left(y\right)\right)\right)$$
11	1 10	bitri	$$⊢F:A\underset{1-1}{⟶}B\leftrightarrow \left(F:A⟶B\wedge \forall x\in A\forall y\in A\left(x\ne y\to F\left(x\right)\ne F\left(y\right)\right)\right)$$