Definition df-dfat 44283

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 44280	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5551	. . . 4 class dom 𝐹
5	1, 4	wcel 2110	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4541	. . . . 5 class {𝐴}
7	2, 6	cres 5553	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6374	. . 3 wff Fun (𝐹 ↾ {𝐴})
9	5, 8	wa 399	. 2 wff (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))
10	3, 9	wb 209	1 wff (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))

This definition is referenced by:  dfateq12d  44290  nfdfat  44291  dfdfat2  44292  fundmdfat  44293  dfatprc  44294  dfatelrn  44295  ndmafv  44304  nfunsnafv  44306  afvpcfv0  44310  afvfvn0fveq  44314  afv0nbfvbi  44315  fnbrafvb  44318  afvelrn  44332  afvres  44336  tz6.12-afv  44337  dmfcoafv  44339  afvco2  44340  aovmpt4g  44365  ndmafv2nrn  44386  funressndmafv2rn  44387  nfunsnafv2  44389  dmafv2rnb  44393  afv2res  44403  tz6.12-afv2  44404  dfatbrafv2b  44409  dfatdmfcoafv2  44418  dfatcolem  44419  dfatco  44420  afv2ndeffv0  44424  afv2fvn0fveq  44428