Definition df-dfat 48015

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 48012	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5659	. . . 4 class dom 𝐹
5	1, 4	wcel 2145	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4587	. . . . 5 class {𝐴}
7	2, 6	cres 5661	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6531	. . 3 wff Fun (𝐹 ↾ {𝐴})
9	5, 8	wa 401	. 2 wff (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))
10	3, 9	wb 209	1 wff (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))

This definition is used by:  dfateq12d  48022  nfdfat  48023  dfdfat2  48024  fundmdfat  48025  dfatprc  48026  dfatelrn  48027  ndmafv  48036  nfunsnafv  48038  afvpcfv0  48042  afvfvn0fveq  48046  afv0nbfvbi  48047  fnbrafvb  48050  afvelrn  48064  afvres  48068  tz6.12-afv  48069  dmfcoafv  48071  afvco2  48072  aovmpt4g  48097  ndmafv2nrn  48118  funressndmafv2rn  48119  nfunsnafv2  48121  dmafv2rnb  48125  afv2res  48135  tz6.12-afv2  48136  dfatbrafv2b  48141  dfatdmfcoafv2  48150  dfatcolem  48151  dfatco  48152  afv2ndeffv0  48156  afv2fvn0fveq  48160