Theorem dmfcoafv

Description: Domains of a function composition, analogous to dmfco . (Contributed by Alexander van der Vekens, 23-Jul-2017)

	Ref	Expression
Assertion	dmfcoafv	⊢ ( ( Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺 ) → ( 𝐴 ∈ dom ( 𝐹 ∘ 𝐺 ) ↔ ( 𝐺 ''' 𝐴 ) ∈ dom 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		dmfco	⊢ ( ( Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺 ) → ( 𝐴 ∈ dom ( 𝐹 ∘ 𝐺 ) ↔ ( 𝐺 ‘ 𝐴 ) ∈ dom 𝐹 ) )
2		funres	⊢ ( Fun 𝐺 → Fun ( 𝐺 ↾ { 𝐴 } ) )
3	2	anim2i	⊢ ( ( 𝐴 ∈ dom 𝐺 ∧ Fun 𝐺 ) → ( 𝐴 ∈ dom 𝐺 ∧ Fun ( 𝐺 ↾ { 𝐴 } ) ) )
4	3	ancoms	⊢ ( ( Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺 ) → ( 𝐴 ∈ dom 𝐺 ∧ Fun ( 𝐺 ↾ { 𝐴 } ) ) )
5		df-dfat	⊢ ( 𝐺 defAt 𝐴 ↔ ( 𝐴 ∈ dom 𝐺 ∧ Fun ( 𝐺 ↾ { 𝐴 } ) ) )
6		afvfundmfveq	⊢ ( 𝐺 defAt 𝐴 → ( 𝐺 ''' 𝐴 ) = ( 𝐺 ‘ 𝐴 ) )
7	5 6	sylbir	⊢ ( ( 𝐴 ∈ dom 𝐺 ∧ Fun ( 𝐺 ↾ { 𝐴 } ) ) → ( 𝐺 ''' 𝐴 ) = ( 𝐺 ‘ 𝐴 ) )
8	4 7	syl	⊢ ( ( Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺 ) → ( 𝐺 ''' 𝐴 ) = ( 𝐺 ‘ 𝐴 ) )
9	8	eqcomd	⊢ ( ( Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺 ) → ( 𝐺 ‘ 𝐴 ) = ( 𝐺 ''' 𝐴 ) )
10	9	eleq1d	⊢ ( ( Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺 ) → ( ( 𝐺 ‘ 𝐴 ) ∈ dom 𝐹 ↔ ( 𝐺 ''' 𝐴 ) ∈ dom 𝐹 ) )
11	1 10	bitrd	⊢ ( ( Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺 ) → ( 𝐴 ∈ dom ( 𝐹 ∘ 𝐺 ) ↔ ( 𝐺 ''' 𝐴 ) ∈ dom 𝐹 ) )