Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > df-dfat | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
df-dfat | ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | cF | . . 3 class 𝐹 | |
3 | 1, 2 | wdfat 44280 | . 2 wff 𝐹 defAt 𝐴 |
4 | 2 | cdm 5551 | . . . 4 class dom 𝐹 |
5 | 1, 4 | wcel 2110 | . . 3 wff 𝐴 ∈ dom 𝐹 |
6 | 1 | csn 4541 | . . . . 5 class {𝐴} |
7 | 2, 6 | cres 5553 | . . . 4 class (𝐹 ↾ {𝐴}) |
8 | 7 | wfun 6374 | . . 3 wff Fun (𝐹 ↾ {𝐴}) |
9 | 5, 8 | wa 399 | . 2 wff (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) |
10 | 3, 9 | wb 209 | 1 wff (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) |
Colors of variables: wff setvar class |
This definition is referenced by: dfateq12d 44290 nfdfat 44291 dfdfat2 44292 fundmdfat 44293 dfatprc 44294 dfatelrn 44295 ndmafv 44304 nfunsnafv 44306 afvpcfv0 44310 afvfvn0fveq 44314 afv0nbfvbi 44315 fnbrafvb 44318 afvelrn 44332 afvres 44336 tz6.12-afv 44337 dmfcoafv 44339 afvco2 44340 aovmpt4g 44365 ndmafv2nrn 44386 funressndmafv2rn 44387 nfunsnafv2 44389 dmafv2rnb 44393 afv2res 44403 tz6.12-afv2 44404 dfatbrafv2b 44409 dfatdmfcoafv2 44418 dfatcolem 44419 dfatco 44420 afv2ndeffv0 44424 afv2fvn0fveq 44428 |
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