Definition df-dfat 41196

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 41193	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5114	. . . 4 class dom 𝐹
5	1, 4	wcel 1990	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4177	. . . . 5 class {𝐴}
7	2, 6	cres 5116	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 5882	. . 3 wff Fun (𝐹 ↾ {𝐴})
9	5, 8	wa 384	. 2 wff (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))
10	3, 9	wb 196	1 wff (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))

This definition is referenced by:  dfateq12d  41209  nfdfat  41210  dfdfat2  41211  ndmafv  41220  nfunsnafv  41222  afvpcfv0  41226  afvfvn0fveq  41230  afv0nbfvbi  41231  fnbrafvb  41234  afvelrn  41248  afvres  41252  tz6.12-afv  41253  dmfcoafv  41255  afvco2  41256  aovmpt4g  41281