Definition df-dfat 47561

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 47558	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5631	. . . 4 class dom 𝐹
5	1, 4	wcel 2114	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4568	. . . . 5 class {𝐴}
7	2, 6	cres 5633	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6493	. . 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  47568  nfdfat  47569  dfdfat2  47570  fundmdfat  47571  dfatprc  47572  dfatelrn  47573  ndmafv  47582  nfunsnafv  47584  afvpcfv0  47588  afvfvn0fveq  47592  afv0nbfvbi  47593  fnbrafvb  47596  afvelrn  47610  afvres  47614  tz6.12-afv  47615  dmfcoafv  47617  afvco2  47618  aovmpt4g  47643  ndmafv2nrn  47664  funressndmafv2rn  47665  nfunsnafv2  47667  dmafv2rnb  47671  afv2res  47681  tz6.12-afv2  47682  dfatbrafv2b  47687  dfatdmfcoafv2  47696  dfatcolem  47697  dfatco  47698  afv2ndeffv0  47702  afv2fvn0fveq  47706