Definition df-dfat 45817

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 45814	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5676	. . . 4 class dom 𝐹
5	1, 4	wcel 2106	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4628	. . . . 5 class {𝐴}
7	2, 6	cres 5678	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6537	. . 3 wff Fun (𝐹 ↾ {𝐴})
9	5, 8	wa 396	. 2 wff (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))
10	3, 9	wb 205	1 wff (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))

This definition is referenced by:  dfateq12d  45824  nfdfat  45825  dfdfat2  45826  fundmdfat  45827  dfatprc  45828  dfatelrn  45829  ndmafv  45838  nfunsnafv  45840  afvpcfv0  45844  afvfvn0fveq  45848  afv0nbfvbi  45849  fnbrafvb  45852  afvelrn  45866  afvres  45870  tz6.12-afv  45871  dmfcoafv  45873  afvco2  45874  aovmpt4g  45899  ndmafv2nrn  45920  funressndmafv2rn  45921  nfunsnafv2  45923  dmafv2rnb  45927  afv2res  45937  tz6.12-afv2  45938  dfatbrafv2b  45943  dfatdmfcoafv2  45952  dfatcolem  45953  dfatco  45954  afv2ndeffv0  45958  afv2fvn0fveq  45962