Theorem dfatafv2rnb

Description: The alternate function value at a class A is defined, i.e. in the range of the function, iff the function is defined at A . (Contributed by AV, 2-Sep-2022)

	Ref	Expression
Assertion	dfatafv2rnb	⊢ ( 𝐹 defAt 𝐴 ↔ ( 𝐹 '''' 𝐴 ) ∈ ran 𝐹 )

Proof

Step	Hyp	Ref	Expression
1		funressndmafv2rn	⊢ ( 𝐹 defAt 𝐴 → ( 𝐹 '''' 𝐴 ) ∈ ran 𝐹 )
2		ndfatafv2nrn	⊢ ( ¬ 𝐹 defAt 𝐴 → ( 𝐹 '''' 𝐴 ) ∉ ran 𝐹 )
3		df-nel	⊢ ( ( 𝐹 '''' 𝐴 ) ∉ ran 𝐹 ↔ ¬ ( 𝐹 '''' 𝐴 ) ∈ ran 𝐹 )
4	2 3	sylib	⊢ ( ¬ 𝐹 defAt 𝐴 → ¬ ( 𝐹 '''' 𝐴 ) ∈ ran 𝐹 )
5	4	con4i	⊢ ( ( 𝐹 '''' 𝐴 ) ∈ ran 𝐹 → 𝐹 defAt 𝐴 )
6	1 5	impbii	⊢ ( 𝐹 defAt 𝐴 ↔ ( 𝐹 '''' 𝐴 ) ∈ ran 𝐹 )