Definition df-dfat 44622

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 44619	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5590	. . . 4 class dom 𝐹
5	1, 4	wcel 2107	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4562	. . . . 5 class {𝐴}
7	2, 6	cres 5592	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6431	. . 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  44629  nfdfat  44630  dfdfat2  44631  fundmdfat  44632  dfatprc  44633  dfatelrn  44634  ndmafv  44643  nfunsnafv  44645  afvpcfv0  44649  afvfvn0fveq  44653  afv0nbfvbi  44654  fnbrafvb  44657  afvelrn  44671  afvres  44675  tz6.12-afv  44676  dmfcoafv  44678  afvco2  44679  aovmpt4g  44704  ndmafv2nrn  44725  funressndmafv2rn  44726  nfunsnafv2  44728  dmafv2rnb  44732  afv2res  44742  tz6.12-afv2  44743  dfatbrafv2b  44748  dfatdmfcoafv2  44757  dfatcolem  44758  dfatco  44759  afv2ndeffv0  44763  afv2fvn0fveq  44767