Theorem foelcdmi

Description: A member of a surjective function's codomain is a value of the function. (Contributed by Thierry Arnoux, 23-Jan-2020)

	Ref	Expression
Assertion	foelcdmi	$$⊢\left(F:A\underset{onto}{⟶}B\wedge Y\in B\right)\to \exists x\in AF\left(x\right)=Y$$

Proof

Step	Hyp	Ref	Expression
1		forn	$$⊢F:A\underset{onto}{⟶}B\to \mathrm{ran}F=B$$
2	1	eleq2d	$$⊢F:A\underset{onto}{⟶}B\to \left(Y\in \mathrm{ran}F\leftrightarrow Y\in B\right)$$
3		fofn	$$⊢F:A\underset{onto}{⟶}B\to FFnA$$
4		fvelrnb	$$⊢FFnA\to \left(Y\in \mathrm{ran}F\leftrightarrow \exists x\in AF\left(x\right)=Y\right)$$
5	3 4	syl	$$⊢F:A\underset{onto}{⟶}B\to \left(Y\in \mathrm{ran}F\leftrightarrow \exists x\in AF\left(x\right)=Y\right)$$
6	2 5	bitr3d	$$⊢F:A\underset{onto}{⟶}B\to \left(Y\in B\leftrightarrow \exists x\in AF\left(x\right)=Y\right)$$
7	6	biimpa	$$⊢\left(F:A\underset{onto}{⟶}B\wedge Y\in B\right)\to \exists x\in AF\left(x\right)=Y$$