Theorem foelrnf

Description: Property of a surjective function. As foelrn but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypothesis	foelrnf.1	$$⊢\underset{\_}{Ⅎ}xF$$
Assertion	foelrnf	$$⊢\left(F:A\underset{onto}{⟶}B\wedge C\in B\right)\to \exists x\in AC=F\left(x\right)$$

Proof

Step	Hyp	Ref	Expression
1		foelrnf.1	$$⊢\underset{\_}{Ⅎ}xF$$
2	1	dffo3f	$$⊢F:A\underset{onto}{⟶}B\leftrightarrow \left(F:A⟶B\wedge \forall y\in B\exists x\in Ay=F\left(x\right)\right)$$
3	2	simprbi	$$⊢F:A\underset{onto}{⟶}B\to \forall y\in B\exists x\in Ay=F\left(x\right)$$
4		eqeq1	$$⊢y=C\to \left(y=F\left(x\right)\leftrightarrow C=F\left(x\right)\right)$$
5	4	rexbidv	$$⊢y=C\to \left(\exists x\in Ay=F\left(x\right)\leftrightarrow \exists x\in AC=F\left(x\right)\right)$$
6	5	rspccva	$$⊢\left(\forall y\in B\exists x\in Ay=F\left(x\right)\wedge C\in B\right)\to \exists x\in AC=F\left(x\right)$$
7	3 6	sylan	$$⊢\left(F:A\underset{onto}{⟶}B\wedge C\in B\right)\to \exists x\in AC=F\left(x\right)$$