Definition df-dfat 47564

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 47561	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5622	. . . 4 class dom 𝐹
5	1, 4	wcel 2114	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4568	. . . . 5 class {𝐴}
7	2, 6	cres 5624	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6484	. . 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  47571  nfdfat  47572  dfdfat2  47573  fundmdfat  47574  dfatprc  47575  dfatelrn  47576  ndmafv  47585  nfunsnafv  47587  afvpcfv0  47591  afvfvn0fveq  47595  afv0nbfvbi  47596  fnbrafvb  47599  afvelrn  47613  afvres  47617  tz6.12-afv  47618  dmfcoafv  47620  afvco2  47621  aovmpt4g  47646  ndmafv2nrn  47667  funressndmafv2rn  47668  nfunsnafv2  47670  dmafv2rnb  47674  afv2res  47684  tz6.12-afv2  47685  dfatbrafv2b  47690  dfatdmfcoafv2  47699  dfatcolem  47700  dfatco  47701  afv2ndeffv0  47705  afv2fvn0fveq  47709