Definition df-dfat 48282

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 48279	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5650	. . . 4 class dom 𝐹
5	1, 4	wcel 2144	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4584	. . . . 5 class {𝐴}
7	2, 6	cres 5652	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6532	. . 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  48289  nfdfat  48290  dfdfat2  48291  fundmdfat  48292  dfatprc  48293  dfatelrn  48294  ndmafv  48303  nfunsnafv  48305  afvpcfv0  48309  afvfvn0fveq  48313  afv0nbfvbi  48314  fnbrafvb  48317  afvelrn  48331  afvres  48335  tz6.12-afv  48336  dmfcoafv  48338  afvco2  48339  aovmpt4g  48364  ndmafv2nrn  48385  funressndmafv2rn  48386  nfunsnafv2  48388  dmafv2rnb  48392  afv2res  48402  tz6.12-afv2  48403  dfatbrafv2b  48408  dfatdmfcoafv2  48417  dfatcolem  48418  dfatco  48419  afv2ndeffv0  48423  afv2fvn0fveq  48427