Definition df-dfat 48395

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 48392	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5655	. . . 4 class dom 𝐹
5	1, 4	wcel 2147	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4588	. . . . 5 class {𝐴}
7	2, 6	cres 5657	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6552	. . 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  48401  nfdfat  48402  dfdfat2  48403  fundmdfat  48404  dfatprc  48405  dfatelrn  48406  ndmafv  48415  nfunsnafv  48417  afvpcfv0  48421  afvfvn0fveq  48425  afv0nbfvbi  48426  fnbrafvb  48429  afvelrn  48443  afvres  48447  tz6.12-afv  48448  dmfcoafv  48450  afvco2  48451  aovmpt4g  48476  ndmafv2nrn  48497  funressndmafv2rn  48498  nfunsnafv2  48500  dmafv2rnb  48504  afv2res  48514  tz6.12-afv2  48515  dfatbrafv2b  48520  dfatdmfcoafv2  48529  dfatcolem  48530  dfatco  48531  afv2ndeffv0  48535  afv2fvn0fveq  48539