Definition df-dfat 47823

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 47820	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5661	. . . 4 class dom 𝐹
5	1, 4	wcel 2141	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4588	. . . . 5 class {𝐴}
7	2, 6	cres 5663	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6530	. . 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  47830  nfdfat  47831  dfdfat2  47832  fundmdfat  47833  dfatprc  47834  dfatelrn  47835  ndmafv  47844  nfunsnafv  47846  afvpcfv0  47850  afvfvn0fveq  47854  afv0nbfvbi  47855  fnbrafvb  47858  afvelrn  47872  afvres  47876  tz6.12-afv  47877  dmfcoafv  47879  afvco2  47880  aovmpt4g  47905  ndmafv2nrn  47926  funressndmafv2rn  47927  nfunsnafv2  47929  dmafv2rnb  47933  afv2res  47943  tz6.12-afv2  47944  dfatbrafv2b  47949  dfatdmfcoafv2  47958  dfatcolem  47959  dfatco  47960  afv2ndeffv0  47964  afv2fvn0fveq  47968