Definition df-dfat 47034

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 47031	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5700	. . . 4 class dom 𝐹
5	1, 4	wcel 2108	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4648	. . . . 5 class {𝐴}
7	2, 6	cres 5702	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6567	. . 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  47041  nfdfat  47042  dfdfat2  47043  fundmdfat  47044  dfatprc  47045  dfatelrn  47046  ndmafv  47055  nfunsnafv  47057  afvpcfv0  47061  afvfvn0fveq  47065  afv0nbfvbi  47066  fnbrafvb  47069  afvelrn  47083  afvres  47087  tz6.12-afv  47088  dmfcoafv  47090  afvco2  47091  aovmpt4g  47116  ndmafv2nrn  47137  funressndmafv2rn  47138  nfunsnafv2  47140  dmafv2rnb  47144  afv2res  47154  tz6.12-afv2  47155  dfatbrafv2b  47160  dfatdmfcoafv2  47169  dfatcolem  47170  dfatco  47171  afv2ndeffv0  47175  afv2fvn0fveq  47179