Theorem nf1oconst

Description: A constant function from at least two elements is not bijective. (Contributed by AV, 30-Mar-2024)

	Ref	Expression
Assertion	nf1oconst	$$⊢\left(F:A⟶\left\{B\right\}\wedge \left(X\in A\wedge Y\in A\wedge X\ne Y\right)\right)\to \neg F:A\underset{1-1\; onto}{⟶}C$$

Proof

Step	Hyp	Ref	Expression
1		nf1const	$$⊢\left(F:A⟶\left\{B\right\}\wedge \left(X\in A\wedge Y\in A\wedge X\ne Y\right)\right)\to \neg F:A\underset{1-1}{⟶}C$$
2	1	orcd	$$⊢\left(F:A⟶\left\{B\right\}\wedge \left(X\in A\wedge Y\in A\wedge X\ne Y\right)\right)\to \left(\neg F:A\underset{1-1}{⟶}C\vee \neg F:A\underset{onto}{⟶}C\right)$$
3		ianor	$$⊢\neg \left(F:A\underset{1-1}{⟶}C\wedge F:A\underset{onto}{⟶}C\right)\leftrightarrow \left(\neg F:A\underset{1-1}{⟶}C\vee \neg F:A\underset{onto}{⟶}C\right)$$
4		df-f1o	$$⊢F:A\underset{1-1\; onto}{⟶}C\leftrightarrow \left(F:A\underset{1-1}{⟶}C\wedge F:A\underset{onto}{⟶}C\right)$$
5	3 4	xchnxbir	$$⊢\neg F:A\underset{1-1\; onto}{⟶}C\leftrightarrow \left(\neg F:A\underset{1-1}{⟶}C\vee \neg F:A\underset{onto}{⟶}C\right)$$
6	2 5	sylibr	$$⊢\left(F:A⟶\left\{B\right\}\wedge \left(X\in A\wedge Y\in A\wedge X\ne Y\right)\right)\to \neg F:A\underset{1-1\; onto}{⟶}C$$