Theorem afvelrn

Description: A function's value belongs to its range, analogous to fvelrn . (Contributed by Alexander van der Vekens, 25-May-2017)

	Ref	Expression
Assertion	afvelrn	⊢ ( ( Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹 ) → ( 𝐹 ''' 𝐴 ) ∈ ran 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		funres	⊢ ( Fun 𝐹 → Fun ( 𝐹 ↾ { 𝐴 } ) )
2	1	anim1i	⊢ ( ( Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹 ) → ( Fun ( 𝐹 ↾ { 𝐴 } ) ∧ 𝐴 ∈ dom 𝐹 ) )
3	2	ancomd	⊢ ( ( Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹 ) → ( 𝐴 ∈ dom 𝐹 ∧ Fun ( 𝐹 ↾ { 𝐴 } ) ) )
4		df-dfat	⊢ ( 𝐹 defAt 𝐴 ↔ ( 𝐴 ∈ dom 𝐹 ∧ Fun ( 𝐹 ↾ { 𝐴 } ) ) )
5	3 4	sylibr	⊢ ( ( Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹 ) → 𝐹 defAt 𝐴 )
6		afvfundmfveq	⊢ ( 𝐹 defAt 𝐴 → ( 𝐹 ''' 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )
7	6	eqcomd	⊢ ( 𝐹 defAt 𝐴 → ( 𝐹 ‘ 𝐴 ) = ( 𝐹 ''' 𝐴 ) )
8	5 7	syl	⊢ ( ( Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹 ) → ( 𝐹 ‘ 𝐴 ) = ( 𝐹 ''' 𝐴 ) )
9		fvelrn	⊢ ( ( Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹 ) → ( 𝐹 ‘ 𝐴 ) ∈ ran 𝐹 )
10	8 9	eqeltrrd	⊢ ( ( Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹 ) → ( 𝐹 ''' 𝐴 ) ∈ ran 𝐹 )