Theorem funcnv

Description: The converse of a class is a function iff the class is single-rooted, which means that for any y in the range of A there is at most one x such that x A y . Definition of single-rooted in Enderton p. 43. See funcnv2 for a simpler version. (Contributed by NM, 13-Aug-2004)

	Ref	Expression
Assertion	funcnv	$$⊢\mathrm{Fun}{A}^{-1}\leftrightarrow \forall y\in \mathrm{ran}A{\exists }^{*}xxAy$$

Proof

Step	Hyp	Ref	Expression
1		vex	$$⊢x\in \mathrm{V}$$
2		vex	$$⊢y\in \mathrm{V}$$
3	1 2	brelrn	$$⊢xAy\to y\in \mathrm{ran}A$$
4	3	pm4.71ri	$$⊢xAy\leftrightarrow \left(y\in \mathrm{ran}A\wedge xAy\right)$$
5	4	mobii	$$⊢{\exists }^{*}xxAy\leftrightarrow {\exists }^{*}x\left(y\in \mathrm{ran}A\wedge xAy\right)$$
6		moanimv	$$⊢{\exists }^{*}x\left(y\in \mathrm{ran}A\wedge xAy\right)\leftrightarrow \left(y\in \mathrm{ran}A\to {\exists }^{*}xxAy\right)$$
7	5 6	bitri	$$⊢{\exists }^{*}xxAy\leftrightarrow \left(y\in \mathrm{ran}A\to {\exists }^{*}xxAy\right)$$
8	7	albii	$$⊢\forall y{\exists }^{*}xxAy\leftrightarrow \forall y\left(y\in \mathrm{ran}A\to {\exists }^{*}xxAy\right)$$
9		funcnv2	$$⊢\mathrm{Fun}{A}^{-1}\leftrightarrow \forall y{\exists }^{*}xxAy$$
10		df-ral	$$⊢\forall y\in \mathrm{ran}A{\exists }^{*}xxAy\leftrightarrow \forall y\left(y\in \mathrm{ran}A\to {\exists }^{*}xxAy\right)$$
11	8 9 10	3bitr4i	$$⊢\mathrm{Fun}{A}^{-1}\leftrightarrow \forall y\in \mathrm{ran}A{\exists }^{*}xxAy$$