Definition df-dfat 47124

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 47121	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5641	. . . 4 class dom 𝐹
5	1, 4	wcel 2109	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4592	. . . . 5 class {𝐴}
7	2, 6	cres 5643	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6508	. . 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  47131  nfdfat  47132  dfdfat2  47133  fundmdfat  47134  dfatprc  47135  dfatelrn  47136  ndmafv  47145  nfunsnafv  47147  afvpcfv0  47151  afvfvn0fveq  47155  afv0nbfvbi  47156  fnbrafvb  47159  afvelrn  47173  afvres  47177  tz6.12-afv  47178  dmfcoafv  47180  afvco2  47181  aovmpt4g  47206  ndmafv2nrn  47227  funressndmafv2rn  47228  nfunsnafv2  47230  dmafv2rnb  47234  afv2res  47244  tz6.12-afv2  47245  dfatbrafv2b  47250  dfatdmfcoafv2  47259  dfatcolem  47260  dfatco  47261  afv2ndeffv0  47265  afv2fvn0fveq  47269