Definition df-dfat 47711

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 47708	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5652	. . . 4 class dom 𝐹
5	1, 4	wcel 2145	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4585	. . . . 5 class {𝐴}
7	2, 6	cres 5654	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6519	. . 3 wff Fun (𝐹 ↾ {𝐴})
9	5, 8	wa 400	. 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 referenced by:  dfateq12d  47718  nfdfat  47719  dfdfat2  47720  fundmdfat  47721  dfatprc  47722  dfatelrn  47723  ndmafv  47732  nfunsnafv  47734  afvpcfv0  47738  afvfvn0fveq  47742  afv0nbfvbi  47743  fnbrafvb  47746  afvelrn  47760  afvres  47764  tz6.12-afv  47765  dmfcoafv  47767  afvco2  47768  aovmpt4g  47793  ndmafv2nrn  47814  funressndmafv2rn  47815  nfunsnafv2  47817  dmafv2rnb  47821  afv2res  47831  tz6.12-afv2  47832  dfatbrafv2b  47837  dfatdmfcoafv2  47846  dfatcolem  47847  dfatco  47848  afv2ndeffv0  47852  afv2fvn0fveq  47856