Theorem ffnaov

Description: An operation maps to a class to which all values belong, analogous to ffnov . (Contributed by Alexander van der Vekens, 26-May-2017)

	Ref	Expression
Assertion	ffnaov	$$⊢F:A\times B⟶C\leftrightarrow \left(FFn\left(A\times B\right)\wedge \forall x\in A\forall y\in B((xFy))\in C\right)$$

Proof

Step	Hyp	Ref	Expression
1		ffnafv	$$⊢F:A\times B⟶C\leftrightarrow \left(FFn\left(A\times B\right)\wedge \forall w\in \left(A\times B\right)\left(F\mathrm{\text{'}\text{'}\text{'}}w\right)\in C\right)$$
2		afveq2	$$⊢w=⟨x,y⟩\to \left(F\mathrm{\text{'}\text{'}\text{'}}w\right)=\left(F\mathrm{\text{'}\text{'}\text{'}}⟨x,y⟩\right)$$
3		df-aov	$$⊢((xFy))=\left(F\mathrm{\text{'}\text{'}\text{'}}⟨x,y⟩\right)$$
4	2 3	eqtr4di	$$⊢w=⟨x,y⟩\to \left(F\mathrm{\text{'}\text{'}\text{'}}w\right)=((xFy))$$
5	4	eleq1d	$$⊢w=⟨x,y⟩\to \left(\left(F\mathrm{\text{'}\text{'}\text{'}}w\right)\in C\leftrightarrow ((xFy))\in C\right)$$
6	5	ralxp	$$⊢\forall w\in \left(A\times B\right)\left(F\mathrm{\text{'}\text{'}\text{'}}w\right)\in C\leftrightarrow \forall x\in A\forall y\in B((xFy))\in C$$
7	6	anbi2i	$$⊢\left(FFn\left(A\times B\right)\wedge \forall w\in \left(A\times B\right)\left(F\mathrm{\text{'}\text{'}\text{'}}w\right)\in C\right)\leftrightarrow \left(FFn\left(A\times B\right)\wedge \forall x\in A\forall y\in B((xFy))\in C\right)$$
8	1 7	bitri	$$⊢F:A\times B⟶C\leftrightarrow \left(FFn\left(A\times B\right)\wedge \forall x\in A\forall y\in B((xFy))\in C\right)$$