Detailed syntax breakdown of Definition df-docaN
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | cocaN 41122 | . 2
class
ocA | 
| 2 |  | vk | . . 3
setvar 𝑘 | 
| 3 |  | cvv 3479 | . . 3
class
V | 
| 4 |  | vw | . . . 4
setvar 𝑤 | 
| 5 | 2 | cv 1538 | . . . . 5
class 𝑘 | 
| 6 |  | clh 39987 | . . . . 5
class
LHyp | 
| 7 | 5, 6 | cfv 6560 | . . . 4
class
(LHyp‘𝑘) | 
| 8 |  | vx | . . . . 5
setvar 𝑥 | 
| 9 | 4 | cv 1538 | . . . . . . 7
class 𝑤 | 
| 10 |  | cltrn 40104 | . . . . . . . 8
class
LTrn | 
| 11 | 5, 10 | cfv 6560 | . . . . . . 7
class
(LTrn‘𝑘) | 
| 12 | 9, 11 | cfv 6560 | . . . . . 6
class
((LTrn‘𝑘)‘𝑤) | 
| 13 | 12 | cpw 4599 | . . . . 5
class 𝒫
((LTrn‘𝑘)‘𝑤) | 
| 14 | 8 | cv 1538 | . . . . . . . . . . . . 13
class 𝑥 | 
| 15 |  | vz | . . . . . . . . . . . . . 14
setvar 𝑧 | 
| 16 | 15 | cv 1538 | . . . . . . . . . . . . 13
class 𝑧 | 
| 17 | 14, 16 | wss 3950 | . . . . . . . . . . . 12
wff 𝑥 ⊆ 𝑧 | 
| 18 |  | cdia 41031 | . . . . . . . . . . . . . . 15
class
DIsoA | 
| 19 | 5, 18 | cfv 6560 | . . . . . . . . . . . . . 14
class
(DIsoA‘𝑘) | 
| 20 | 9, 19 | cfv 6560 | . . . . . . . . . . . . 13
class
((DIsoA‘𝑘)‘𝑤) | 
| 21 | 20 | crn 5685 | . . . . . . . . . . . 12
class ran
((DIsoA‘𝑘)‘𝑤) | 
| 22 | 17, 15, 21 | crab 3435 | . . . . . . . . . . 11
class {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧} | 
| 23 | 22 | cint 4945 | . . . . . . . . . 10
class ∩ {𝑧
∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧} | 
| 24 | 20 | ccnv 5683 | . . . . . . . . . 10
class ◡((DIsoA‘𝑘)‘𝑤) | 
| 25 | 23, 24 | cfv 6560 | . . . . . . . . 9
class (◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}) | 
| 26 |  | coc 17306 | . . . . . . . . . 10
class
oc | 
| 27 | 5, 26 | cfv 6560 | . . . . . . . . 9
class
(oc‘𝑘) | 
| 28 | 25, 27 | cfv 6560 | . . . . . . . 8
class
((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) | 
| 29 | 9, 27 | cfv 6560 | . . . . . . . 8
class
((oc‘𝑘)‘𝑤) | 
| 30 |  | cjn 18358 | . . . . . . . . 9
class
join | 
| 31 | 5, 30 | cfv 6560 | . . . . . . . 8
class
(join‘𝑘) | 
| 32 | 28, 29, 31 | co 7432 | . . . . . . 7
class
(((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤)) | 
| 33 |  | cmee 18359 | . . . . . . . 8
class
meet | 
| 34 | 5, 33 | cfv 6560 | . . . . . . 7
class
(meet‘𝑘) | 
| 35 | 32, 9, 34 | co 7432 | . . . . . 6
class
((((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤) | 
| 36 | 35, 20 | cfv 6560 | . . . . 5
class
(((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤)) | 
| 37 | 8, 13, 36 | cmpt 5224 | . . . 4
class (𝑥 ∈ 𝒫
((LTrn‘𝑘)‘𝑤) ↦ (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤))) | 
| 38 | 4, 7, 37 | cmpt 5224 | . . 3
class (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤)))) | 
| 39 | 2, 3, 38 | cmpt 5224 | . 2
class (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤))))) | 
| 40 | 1, 39 | wceq 1539 | 1
wff ocA =
(𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤))))) |