Theorem dff14b

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	dff14b	$$⊢F:A\underset{1-1}{⟶}B\leftrightarrow \left(F:A⟶B\wedge \forall x\in A\forall y\in \left(A\setminus \left\{x\right\}\right)F\left(x\right)\ne F\left(y\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		dff14a	$$⊢F:A\underset{1-1}{⟶}B\leftrightarrow \left(F:A⟶B\wedge \forall x\in A\forall y\in A\left(x\ne y\to F\left(x\right)\ne F\left(y\right)\right)\right)$$
2		necom	$$⊢x\ne y\leftrightarrow y\ne x$$
3	2	imbi1i	$$⊢\left(x\ne y\to F\left(x\right)\ne F\left(y\right)\right)\leftrightarrow \left(y\ne x\to F\left(x\right)\ne F\left(y\right)\right)$$
4	3	ralbii	$$⊢\forall y\in A\left(x\ne y\to F\left(x\right)\ne F\left(y\right)\right)\leftrightarrow \forall y\in A\left(y\ne x\to F\left(x\right)\ne F\left(y\right)\right)$$
5		raldifsnb	$$⊢\forall y\in A\left(y\ne x\to F\left(x\right)\ne F\left(y\right)\right)\leftrightarrow \forall y\in \left(A\setminus \left\{x\right\}\right)F\left(x\right)\ne F\left(y\right)$$
6	4 5	bitri	$$⊢\forall y\in A\left(x\ne y\to F\left(x\right)\ne F\left(y\right)\right)\leftrightarrow \forall y\in \left(A\setminus \left\{x\right\}\right)F\left(x\right)\ne F\left(y\right)$$
7	6	ralbii	$$⊢\forall x\in A\forall y\in A\left(x\ne y\to F\left(x\right)\ne F\left(y\right)\right)\leftrightarrow \forall x\in A\forall y\in \left(A\setminus \left\{x\right\}\right)F\left(x\right)\ne F\left(y\right)$$
8	7	anbi2i	$$⊢\left(F:A⟶B\wedge \forall x\in A\forall y\in A\left(x\ne y\to F\left(x\right)\ne F\left(y\right)\right)\right)\leftrightarrow \left(F:A⟶B\wedge \forall x\in A\forall y\in \left(A\setminus \left\{x\right\}\right)F\left(x\right)\ne F\left(y\right)\right)$$
9	1 8	bitri	$$⊢F:A\underset{1-1}{⟶}B\leftrightarrow \left(F:A⟶B\wedge \forall x\in A\forall y\in \left(A\setminus \left\{x\right\}\right)F\left(x\right)\ne F\left(y\right)\right)$$