Definition df-dfat 47468

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 47465	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5632	. . . 4 class dom 𝐹
5	1, 4	wcel 2114	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4582	. . . . 5 class {𝐴}
7	2, 6	cres 5634	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6494	. . 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  47475  nfdfat  47476  dfdfat2  47477  fundmdfat  47478  dfatprc  47479  dfatelrn  47480  ndmafv  47489  nfunsnafv  47491  afvpcfv0  47495  afvfvn0fveq  47499  afv0nbfvbi  47500  fnbrafvb  47503  afvelrn  47517  afvres  47521  tz6.12-afv  47522  dmfcoafv  47524  afvco2  47525  aovmpt4g  47550  ndmafv2nrn  47571  funressndmafv2rn  47572  nfunsnafv2  47574  dmafv2rnb  47578  afv2res  47588  tz6.12-afv2  47589  dfatbrafv2b  47594  dfatdmfcoafv2  47603  dfatcolem  47604  dfatco  47605  afv2ndeffv0  47609  afv2fvn0fveq  47613