Theorem elrnmptdv

Description: Elementhood in the range of a function in maps-to notation, deduction form. (Contributed by Rohan Ridenour, 3-Aug-2023)

	Ref	Expression
Hypotheses	elrnmptdv.1	$$⊢F=\left(x\in A⟼B\right)$$
	elrnmptdv.2	$$⊢\phi \to C\in A$$
	elrnmptdv.3	$$⊢\phi \to D\in V$$
	elrnmptdv.4	$$⊢\left(\phi \wedge x=C\right)\to D=B$$
Assertion	elrnmptdv	$$⊢\phi \to D\in \mathrm{ran}F$$

Proof

Step	Hyp	Ref	Expression
1		elrnmptdv.1	$$⊢F=\left(x\in A⟼B\right)$$
2		elrnmptdv.2	$$⊢\phi \to C\in A$$
3		elrnmptdv.3	$$⊢\phi \to D\in V$$
4		elrnmptdv.4	$$⊢\left(\phi \wedge x=C\right)\to D=B$$
5	4 2	rspcime	$$⊢\phi \to \exists x\in AD=B$$
6	1	elrnmpt	$$⊢D\in V\to \left(D\in \mathrm{ran}F\leftrightarrow \exists x\in AD=B\right)$$
7	3 6	syl	$$⊢\phi \to \left(D\in \mathrm{ran}F\leftrightarrow \exists x\in AD=B\right)$$
8	5 7	mpbird	$$⊢\phi \to D\in \mathrm{ran}F$$