Definition df-dfat 47148

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 47145	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5654	. . . 4 class dom 𝐹
5	1, 4	wcel 2108	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4601	. . . . 5 class {𝐴}
7	2, 6	cres 5656	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6525	. . 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  47155  nfdfat  47156  dfdfat2  47157  fundmdfat  47158  dfatprc  47159  dfatelrn  47160  ndmafv  47169  nfunsnafv  47171  afvpcfv0  47175  afvfvn0fveq  47179  afv0nbfvbi  47180  fnbrafvb  47183  afvelrn  47197  afvres  47201  tz6.12-afv  47202  dmfcoafv  47204  afvco2  47205  aovmpt4g  47230  ndmafv2nrn  47251  funressndmafv2rn  47252  nfunsnafv2  47254  dmafv2rnb  47258  afv2res  47268  tz6.12-afv2  47269  dfatbrafv2b  47274  dfatdmfcoafv2  47283  dfatcolem  47284  dfatco  47285  afv2ndeffv0  47289  afv2fvn0fveq  47293