Definition df-dfat 47104

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 47101	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5623	. . . 4 class dom 𝐹
5	1, 4	wcel 2109	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4579	. . . . 5 class {𝐴}
7	2, 6	cres 5625	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6480	. . 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  47111  nfdfat  47112  dfdfat2  47113  fundmdfat  47114  dfatprc  47115  dfatelrn  47116  ndmafv  47125  nfunsnafv  47127  afvpcfv0  47131  afvfvn0fveq  47135  afv0nbfvbi  47136  fnbrafvb  47139  afvelrn  47153  afvres  47157  tz6.12-afv  47158  dmfcoafv  47160  afvco2  47161  aovmpt4g  47186  ndmafv2nrn  47207  funressndmafv2rn  47208  nfunsnafv2  47210  dmafv2rnb  47214  afv2res  47224  tz6.12-afv2  47225  dfatbrafv2b  47230  dfatdmfcoafv2  47239  dfatcolem  47240  dfatco  47241  afv2ndeffv0  47245  afv2fvn0fveq  47249