Definition df-dfat 47401

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 47398	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5625	. . . 4 class dom 𝐹
5	1, 4	wcel 2114	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4581	. . . . 5 class {𝐴}
7	2, 6	cres 5627	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6487	. . 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  47408  nfdfat  47409  dfdfat2  47410  fundmdfat  47411  dfatprc  47412  dfatelrn  47413  ndmafv  47422  nfunsnafv  47424  afvpcfv0  47428  afvfvn0fveq  47432  afv0nbfvbi  47433  fnbrafvb  47436  afvelrn  47450  afvres  47454  tz6.12-afv  47455  dmfcoafv  47457  afvco2  47458  aovmpt4g  47483  ndmafv2nrn  47504  funressndmafv2rn  47505  nfunsnafv2  47507  dmafv2rnb  47511  afv2res  47521  tz6.12-afv2  47522  dfatbrafv2b  47527  dfatdmfcoafv2  47536  dfatcolem  47537  dfatco  47538  afv2ndeffv0  47542  afv2fvn0fveq  47546