Theorem funoprabg

Description: "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 28-Aug-2007)

	Ref	Expression
Assertion	funoprabg	$$⊢\forall x\forall y{\exists }^{*}z\phi \to \mathrm{Fun}\left\{⟨⟨x,y⟩,z⟩|\phi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		mosubopt	$$⊢\forall x\forall y{\exists }^{*}z\phi \to {\exists }^{*}z\exists x\exists y\left(w=⟨x,y⟩\wedge \phi \right)$$
2	1	alrimiv	$$⊢\forall x\forall y{\exists }^{*}z\phi \to \forall w{\exists }^{*}z\exists x\exists y\left(w=⟨x,y⟩\wedge \phi \right)$$
3		dfoprab2	$$⊢\left\{⟨⟨x,y⟩,z⟩|\phi \right\}=\left\{⟨w,z⟩|\exists x\exists y\left(w=⟨x,y⟩\wedge \phi \right)\right\}$$
4	3	funeqi	$$⊢\mathrm{Fun}\left\{⟨⟨x,y⟩,z⟩|\phi \right\}\leftrightarrow \mathrm{Fun}\left\{⟨w,z⟩|\exists x\exists y\left(w=⟨x,y⟩\wedge \phi \right)\right\}$$
5		funopab	$$⊢\mathrm{Fun}\left\{⟨w,z⟩|\exists x\exists y\left(w=⟨x,y⟩\wedge \phi \right)\right\}\leftrightarrow \forall w{\exists }^{*}z\exists x\exists y\left(w=⟨x,y⟩\wedge \phi \right)$$
6	4 5	bitr2i	$$⊢\forall w{\exists }^{*}z\exists x\exists y\left(w=⟨x,y⟩\wedge \phi \right)\leftrightarrow \mathrm{Fun}\left\{⟨⟨x,y⟩,z⟩|\phi \right\}$$
7	2 6	sylib	$$⊢\forall x\forall y{\exists }^{*}z\phi \to \mathrm{Fun}\left\{⟨⟨x,y⟩,z⟩|\phi \right\}$$