Definition df-dfat 47129

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 47126	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5614	. . . 4 class dom 𝐹
5	1, 4	wcel 2110	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4574	. . . . 5 class {𝐴}
7	2, 6	cres 5616	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6471	. . 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  47136  nfdfat  47137  dfdfat2  47138  fundmdfat  47139  dfatprc  47140  dfatelrn  47141  ndmafv  47150  nfunsnafv  47152  afvpcfv0  47156  afvfvn0fveq  47160  afv0nbfvbi  47161  fnbrafvb  47164  afvelrn  47178  afvres  47182  tz6.12-afv  47183  dmfcoafv  47185  afvco2  47186  aovmpt4g  47211  ndmafv2nrn  47232  funressndmafv2rn  47233  nfunsnafv2  47235  dmafv2rnb  47239  afv2res  47249  tz6.12-afv2  47250  dfatbrafv2b  47255  dfatdmfcoafv2  47264  dfatcolem  47265  dfatco  47266  afv2ndeffv0  47270  afv2fvn0fveq  47274