Definition df-dfat 47120

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 47117	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5638	. . . 4 class dom 𝐹
5	1, 4	wcel 2109	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4589	. . . . 5 class {𝐴}
7	2, 6	cres 5640	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6505	. . 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  47127  nfdfat  47128  dfdfat2  47129  fundmdfat  47130  dfatprc  47131  dfatelrn  47132  ndmafv  47141  nfunsnafv  47143  afvpcfv0  47147  afvfvn0fveq  47151  afv0nbfvbi  47152  fnbrafvb  47155  afvelrn  47169  afvres  47173  tz6.12-afv  47174  dmfcoafv  47176  afvco2  47177  aovmpt4g  47202  ndmafv2nrn  47223  funressndmafv2rn  47224  nfunsnafv2  47226  dmafv2rnb  47230  afv2res  47240  tz6.12-afv2  47241  dfatbrafv2b  47246  dfatdmfcoafv2  47255  dfatcolem  47256  dfatco  47257  afv2ndeffv0  47261  afv2fvn0fveq  47265