Definition df-dfat 47666

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 47663	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5645	. . . 4 class dom 𝐹
5	1, 4	wcel 2141	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4581	. . . . 5 class {𝐴}
7	2, 6	cres 5647	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6509	. . 3 wff Fun (𝐹 ↾ {𝐴})
9	5, 8	wa 399	. 2 wff (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))
10	3, 9	wb 208	1 wff (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))

This definition is referenced by:  dfateq12d  47673  nfdfat  47674  dfdfat2  47675  fundmdfat  47676  dfatprc  47677  dfatelrn  47678  ndmafv  47687  nfunsnafv  47689  afvpcfv0  47693  afvfvn0fveq  47697  afv0nbfvbi  47698  fnbrafvb  47701  afvelrn  47715  afvres  47719  tz6.12-afv  47720  dmfcoafv  47722  afvco2  47723  aovmpt4g  47748  ndmafv2nrn  47769  funressndmafv2rn  47770  nfunsnafv2  47772  dmafv2rnb  47776  afv2res  47786  tz6.12-afv2  47787  dfatbrafv2b  47792  dfatdmfcoafv2  47801  dfatcolem  47802  dfatco  47803  afv2ndeffv0  47807  afv2fvn0fveq  47811