Definition df-dfat 39845

Description: Definition of the predicate that determines if some class 𝐹 is defined as function for an argument 𝐴 or, in other words, if the function value for some class 𝐹 for an argument 𝐴 is defined. We say that 𝐹 is defined at 𝐴 if a 𝐹 is a function restricted to the member 𝐴 of its domain. (Contributed by Alexander van der Vekens, 25-May-2017.)

Assertion

Ref	Expression
df-dfat	⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))

Detailed syntax breakdown of Definition df-dfat

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cF	. . 3 class 𝐹
3	1, 2	wdfat 39842	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5038	. . . 4 class dom 𝐹
5	1, 4	wcel 1977	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4125	. . . . 5 class {𝐴}
7	2, 6	cres 5040	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 5798	. . 3 wff Fun (𝐹 ↾ {𝐴})
9	5, 8	wa 383	. 2 wff (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))
10	3, 9	wb 195	1 wff (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))

This definition is referenced by:  dfateq12d  39858  nfdfat  39859  dfdfat2  39860  ndmafv  39869  nfunsnafv  39871  afvpcfv0  39875  afvfvn0fveq  39879  afv0nbfvbi  39880  fnbrafvb  39883  afvelrn  39897  afvres  39901  tz6.12-afv  39902  dmfcoafv  39904  afvco2  39905  aovmpt4g  39930