Theorem foelrn

Description: Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011)

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

Proof

Step	Hyp	Ref	Expression
1		dffo3	$$⊢F:A\underset{onto}{⟶}B\leftrightarrow \left(F:A⟶B\wedge \forall y\in B\exists x\in Ay=F\left(x\right)\right)$$
2	1	simprbi	$$⊢F:A\underset{onto}{⟶}B\to \forall y\in B\exists x\in Ay=F\left(x\right)$$
3		eqeq1	$$⊢y=C\to \left(y=F\left(x\right)\leftrightarrow C=F\left(x\right)\right)$$
4	3	rexbidv	$$⊢y=C\to \left(\exists x\in Ay=F\left(x\right)\leftrightarrow \exists x\in AC=F\left(x\right)\right)$$
5	4	rspccva	$$⊢\left(\forall y\in B\exists x\in Ay=F\left(x\right)\wedge C\in B\right)\to \exists x\in AC=F\left(x\right)$$
6	2 5	sylan	$$⊢\left(F:A\underset{onto}{⟶}B\wedge C\in B\right)\to \exists x\in AC=F\left(x\right)$$