Definition df-dfat 43312

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 43309	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5549	. . . 4 class dom 𝐹
5	1, 4	wcel 2110	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4560	. . . . 5 class {𝐴}
7	2, 6	cres 5551	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6343	. . 3 wff Fun (𝐹 ↾ {𝐴})
9	5, 8	wa 398	. 2 wff (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))
10	3, 9	wb 208	1 wff (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))

This definition is referenced by:  dfateq12d  43319  nfdfat  43320  dfdfat2  43321  fundmdfat  43322  dfatprc  43323  dfatelrn  43324  ndmafv  43333  nfunsnafv  43335  afvpcfv0  43339  afvfvn0fveq  43343  afv0nbfvbi  43344  fnbrafvb  43347  afvelrn  43361  afvres  43365  tz6.12-afv  43366  dmfcoafv  43368  afvco2  43369  aovmpt4g  43394  ndmafv2nrn  43415  funressndmafv2rn  43416  nfunsnafv2  43418  dmafv2rnb  43422  afv2res  43432  tz6.12-afv2  43433  dfatbrafv2b  43438  dfatdmfcoafv2  43447  dfatcolem  43448  dfatco  43449  afv2ndeffv0  43453  afv2fvn0fveq  43457