Definition df-dfat 47889

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 47886	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5666	. . . 4 class dom 𝐹
5	1, 4	wcel 2146	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4594	. . . . 5 class {𝐴}
7	2, 6	cres 5668	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6537	. . 3 wff Fun (𝐹 ↾ {𝐴})
9	5, 8	wa 401	. 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 used by:  dfateq12d  47896  nfdfat  47897  dfdfat2  47898  fundmdfat  47899  dfatprc  47900  dfatelrn  47901  ndmafv  47910  nfunsnafv  47912  afvpcfv0  47916  afvfvn0fveq  47920  afv0nbfvbi  47921  fnbrafvb  47924  afvelrn  47938  afvres  47942  tz6.12-afv  47943  dmfcoafv  47945  afvco2  47946  aovmpt4g  47971  ndmafv2nrn  47992  funressndmafv2rn  47993  nfunsnafv2  47995  dmafv2rnb  47999  afv2res  48009  tz6.12-afv2  48010  dfatbrafv2b  48015  dfatdmfcoafv2  48024  dfatcolem  48025  dfatco  48026  afv2ndeffv0  48030  afv2fvn0fveq  48034