Definition df-dfat 47233

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 47230	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5621	. . . 4 class dom 𝐹
5	1, 4	wcel 2113	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4577	. . . . 5 class {𝐴}
7	2, 6	cres 5623	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6483	. . 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  47240  nfdfat  47241  dfdfat2  47242  fundmdfat  47243  dfatprc  47244  dfatelrn  47245  ndmafv  47254  nfunsnafv  47256  afvpcfv0  47260  afvfvn0fveq  47264  afv0nbfvbi  47265  fnbrafvb  47268  afvelrn  47282  afvres  47286  tz6.12-afv  47287  dmfcoafv  47289  afvco2  47290  aovmpt4g  47315  ndmafv2nrn  47336  funressndmafv2rn  47337  nfunsnafv2  47339  dmafv2rnb  47343  afv2res  47353  tz6.12-afv2  47354  dfatbrafv2b  47359  dfatdmfcoafv2  47368  dfatcolem  47369  dfatco  47370  afv2ndeffv0  47374  afv2fvn0fveq  47378