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 = ∪ 𝑥 ∈ 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 ∈ dom 𝐹 ↦ ( 𝐹 ‘ 𝑥 ) ) )
3	1 2	sylbb	⊢ ( Fun 𝐹 → 𝐹 = ( 𝑥 ∈ dom 𝐹 ↦ ( 𝐹 ‘ 𝑥 ) ) )
4		fvex	⊢ ( 𝐹 ‘ 𝑥 ) ∈ V
5	4	dfmpt	⊢ ( 𝑥 ∈ dom 𝐹 ↦ ( 𝐹 ‘ 𝑥 ) ) = ∪ 𝑥 ∈ dom 𝐹 { ⟨ 𝑥 , ( 𝐹 ‘ 𝑥 ) ⟩ }
6	3 5	eqtrdi	⊢ ( Fun 𝐹 → 𝐹 = ∪ 𝑥 ∈ dom 𝐹 { ⟨ 𝑥 , ( 𝐹 ‘ 𝑥 ) ⟩ } )