Theorem funiun

Description: A function is a union of singletons of ordered pairs indexed by its domain. (Contributed by AV, 18-Sep-2020)

	Ref	Expression
Assertion	funiun	|- ( Fun F -> F = U_ x e. dom F { <. x , ( F ` x ) >. } )

Proof

Step	Hyp	Ref	Expression
1		funfn	|- ( Fun F <-> F Fn dom F )
2		dffn5	|- ( F Fn dom F <-> F = ( x e. dom F |-> ( F ` x ) ) )
3	1 2	sylbb	|- ( Fun F -> F = ( x e. dom F |-> ( F ` x ) ) )
4		fvex	|- ( F ` x ) e. _V
5	4	dfmpt	|- ( x e. dom F |-> ( F ` x ) ) = U_ x e. dom F { <. x , ( F ` x ) >. }
6	3 5	eqtrdi	|- ( Fun F -> F = U_ x e. dom F { <. x , ( F ` x ) >. } )