Definition df-dfat 44498

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 44495	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5580	. . . 4 class dom 𝐹
5	1, 4	wcel 2108	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4558	. . . . 5 class {𝐴}
7	2, 6	cres 5582	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6412	. . 3 wff Fun (𝐹 ↾ {𝐴})
9	5, 8	wa 395	. 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  44505  nfdfat  44506  dfdfat2  44507  fundmdfat  44508  dfatprc  44509  dfatelrn  44510  ndmafv  44519  nfunsnafv  44521  afvpcfv0  44525  afvfvn0fveq  44529  afv0nbfvbi  44530  fnbrafvb  44533  afvelrn  44547  afvres  44551  tz6.12-afv  44552  dmfcoafv  44554  afvco2  44555  aovmpt4g  44580  ndmafv2nrn  44601  funressndmafv2rn  44602  nfunsnafv2  44604  dmafv2rnb  44608  afv2res  44618  tz6.12-afv2  44619  dfatbrafv2b  44624  dfatdmfcoafv2  44633  dfatcolem  44634  dfatco  44635  afv2ndeffv0  44639  afv2fvn0fveq  44643