Theorem fnoprab

Description: Functionality and domain of an operation class abstraction. (Contributed by NM, 15-May-1995)

	Ref	Expression
Hypothesis	fnoprab.1	$$⊢\phi \to \exists !z\psi$$
Assertion	fnoprab	$$⊢\left\{⟨⟨x,y⟩,z⟩|\left(\phi \wedge \psi \right)\right\}Fn\left\{⟨x,y⟩|\phi \right\}$$

Proof

Step	Hyp	Ref	Expression
1		fnoprab.1	$$⊢\phi \to \exists !z\psi$$
2	1	gen2	$$⊢\forall x\forall y\left(\phi \to \exists !z\psi \right)$$
3		fnoprabg	$$⊢\forall x\forall y\left(\phi \to \exists !z\psi \right)\to \left\{⟨⟨x,y⟩,z⟩|\left(\phi \wedge \psi \right)\right\}Fn\left\{⟨x,y⟩|\phi \right\}$$
4	2 3	ax-mp	$$⊢\left\{⟨⟨x,y⟩,z⟩|\left(\phi \wedge \psi \right)\right\}Fn\left\{⟨x,y⟩|\phi \right\}$$