Definition df-dfat 46126

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 46123	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5676	. . . 4 class dom 𝐹
5	1, 4	wcel 2106	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4628	. . . . 5 class {𝐴}
7	2, 6	cres 5678	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6537	. . 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  46133  nfdfat  46134  dfdfat2  46135  fundmdfat  46136  dfatprc  46137  dfatelrn  46138  ndmafv  46147  nfunsnafv  46149  afvpcfv0  46153  afvfvn0fveq  46157  afv0nbfvbi  46158  fnbrafvb  46161  afvelrn  46175  afvres  46179  tz6.12-afv  46180  dmfcoafv  46182  afvco2  46183  aovmpt4g  46208  ndmafv2nrn  46229  funressndmafv2rn  46230  nfunsnafv2  46232  dmafv2rnb  46236  afv2res  46246  tz6.12-afv2  46247  dfatbrafv2b  46252  dfatdmfcoafv2  46261  dfatcolem  46262  dfatco  46263  afv2ndeffv0  46267  afv2fvn0fveq  46271