Definition df-dfat 48130

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 48127	. 2 wff 𝐹 defAt 𝐴
4	2	cdm 5651	. . . 4 class dom 𝐹
5	1, 4	wcel 2145	. . 3 wff 𝐴 ∈ dom 𝐹
6	1	csn 4584	. . . . 5 class {𝐴}
7	2, 6	cres 5653	. . . 4 class (𝐹 ↾ {𝐴})
8	7	wfun 6525	. . 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  48137  nfdfat  48138  dfdfat2  48139  fundmdfat  48140  dfatprc  48141  dfatelrn  48142  ndmafv  48151  nfunsnafv  48153  afvpcfv0  48157  afvfvn0fveq  48161  afv0nbfvbi  48162  fnbrafvb  48165  afvelrn  48179  afvres  48183  tz6.12-afv  48184  dmfcoafv  48186  afvco2  48187  aovmpt4g  48212  ndmafv2nrn  48233  funressndmafv2rn  48234  nfunsnafv2  48236  dmafv2rnb  48240  afv2res  48250  tz6.12-afv2  48251  dfatbrafv2b  48256  dfatdmfcoafv2  48265  dfatcolem  48266  dfatco  48267  afv2ndeffv0  48271  afv2fvn0fveq  48275