Definition df-dfat 47131

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 47128	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5685	. . . 4 class dom 𝐹
5	1, 4	wcel 2108	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4626	. . . . 5 class {𝐴}
7	2, 6	cres 5687	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6555	. . 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  47138  nfdfat  47139  dfdfat2  47140  fundmdfat  47141  dfatprc  47142  dfatelrn  47143  ndmafv  47152  nfunsnafv  47154  afvpcfv0  47158  afvfvn0fveq  47162  afv0nbfvbi  47163  fnbrafvb  47166  afvelrn  47180  afvres  47184  tz6.12-afv  47185  dmfcoafv  47187  afvco2  47188  aovmpt4g  47213  ndmafv2nrn  47234  funressndmafv2rn  47235  nfunsnafv2  47237  dmafv2rnb  47241  afv2res  47251  tz6.12-afv2  47252  dfatbrafv2b  47257  dfatdmfcoafv2  47266  dfatcolem  47267  dfatco  47268  afv2ndeffv0  47272  afv2fvn0fveq  47276