Definition df-dfat 42834

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 42831	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5443	. . . 4 class dom 𝐹
5	1, 4	wcel 2081	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4472	. . . . 5 class {𝐴}
7	2, 6	cres 5445	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6219	. . 3 wff Fun (𝐹 ↾ {𝐴})
9	5, 8	wa 396	. 2 wff (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))
10	3, 9	wb 207	1 wff (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))

This definition is referenced by:  dfateq12d  42841  nfdfat  42842  dfdfat2  42843  fundmdfat  42844  dfatprc  42845  dfatelrn  42846  ndmafv  42855  nfunsnafv  42857  afvpcfv0  42861  afvfvn0fveq  42865  afv0nbfvbi  42866  fnbrafvb  42869  afvelrn  42883  afvres  42887  tz6.12-afv  42888  dmfcoafv  42890  afvco2  42891  aovmpt4g  42916  ndmafv2nrn  42937  funressndmafv2rn  42938  nfunsnafv2  42940  dmafv2rnb  42944  afv2res  42954  tz6.12-afv2  42955  dfatbrafv2b  42960  dfatdmfcoafv2  42969  dfatcolem  42970  dfatco  42971  afv2ndeffv0  42975  afv2fvn0fveq  42979