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 43192 | . 2 wff 𝐹 defAt 𝐴 |
4 | 2 | cdm 5548 | . . . 4 class dom 𝐹 |
5 | 1, 4 | wcel 2105 | . . 3 wff 𝐴 ∈ dom 𝐹 |
6 | 1 | csn 4557 | . . . . 5 class {𝐴} |
7 | 2, 6 | cres 5550 | . . . 4 class (𝐹 ↾ {𝐴}) |
8 | 7 | wfun 6342 | . . 3 wff Fun (𝐹 ↾ {𝐴}) |
9 | 5, 8 | wa 396 | . 2 wff (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) |
10 | 3, 9 | wb 207 | 1 wff (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) |
Colors of variables: wff setvar class |
This definition is referenced by: dfateq12d 43202 nfdfat 43203 dfdfat2 43204 fundmdfat 43205 dfatprc 43206 dfatelrn 43207 ndmafv 43216 nfunsnafv 43218 afvpcfv0 43222 afvfvn0fveq 43226 afv0nbfvbi 43227 fnbrafvb 43230 afvelrn 43244 afvres 43248 tz6.12-afv 43249 dmfcoafv 43251 afvco2 43252 aovmpt4g 43277 ndmafv2nrn 43298 funressndmafv2rn 43299 nfunsnafv2 43301 dmafv2rnb 43305 afv2res 43315 tz6.12-afv2 43316 dfatbrafv2b 43321 dfatdmfcoafv2 43330 dfatcolem 43331 dfatco 43332 afv2ndeffv0 43336 afv2fvn0fveq 43340 |
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