Definition df-dfat 47069

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 47066	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5689	. . . 4 class dom 𝐹
5	1, 4	wcel 2106	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4631	. . . . 5 class {𝐴}
7	2, 6	cres 5691	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6557	. . 3 wff Fun (𝐹 ↾ {𝐴})
9	5, 8	wa 395	. 2 wff (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))
10	3, 9	wb 206	1 wff (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))

This definition is referenced by:  dfateq12d  47076  nfdfat  47077  dfdfat2  47078  fundmdfat  47079  dfatprc  47080  dfatelrn  47081  ndmafv  47090  nfunsnafv  47092  afvpcfv0  47096  afvfvn0fveq  47100  afv0nbfvbi  47101  fnbrafvb  47104  afvelrn  47118  afvres  47122  tz6.12-afv  47123  dmfcoafv  47125  afvco2  47126  aovmpt4g  47151  ndmafv2nrn  47172  funressndmafv2rn  47173  nfunsnafv2  47175  dmafv2rnb  47179  afv2res  47189  tz6.12-afv2  47190  dfatbrafv2b  47195  dfatdmfcoafv2  47204  dfatcolem  47205  dfatco  47206  afv2ndeffv0  47210  afv2fvn0fveq  47214