Definition df-dfat 47744

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 47741	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5662	. . . 4 class dom 𝐹
5	1, 4	wcel 2149	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4594	. . . . 5 class {𝐴}
7	2, 6	cres 5664	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6531	. . 3 wff Fun (𝐹 ↾ {𝐴})
9	5, 8	wa 400	. 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  47751  nfdfat  47752  dfdfat2  47753  fundmdfat  47754  dfatprc  47755  dfatelrn  47756  ndmafv  47765  nfunsnafv  47767  afvpcfv0  47771  afvfvn0fveq  47775  afv0nbfvbi  47776  fnbrafvb  47779  afvelrn  47793  afvres  47797  tz6.12-afv  47798  dmfcoafv  47800  afvco2  47801  aovmpt4g  47826  ndmafv2nrn  47847  funressndmafv2rn  47848  nfunsnafv2  47850  dmafv2rnb  47854  afv2res  47864  tz6.12-afv2  47865  dfatbrafv2b  47870  dfatdmfcoafv2  47879  dfatcolem  47880  dfatco  47881  afv2ndeffv0  47885  afv2fvn0fveq  47889