Detailed syntax breakdown of Definition df-grtri
Step | Hyp | Ref
| Expression |
1 | | cgrtri 47718 |
. 2
class
GrTriangles |
2 | | vg |
. . 3
setvar 𝑔 |
3 | | cvv 3482 |
. . 3
class
V |
4 | | vv |
. . . 4
setvar 𝑣 |
5 | 2 | cv 1536 |
. . . . 5
class 𝑔 |
6 | | cvtx 29022 |
. . . . 5
class
Vtx |
7 | 5, 6 | cfv 6572 |
. . . 4
class
(Vtx‘𝑔) |
8 | | ve |
. . . . 5
setvar 𝑒 |
9 | | cedg 29073 |
. . . . . 6
class
Edg |
10 | 5, 9 | cfv 6572 |
. . . . 5
class
(Edg‘𝑔) |
11 | | cc0 11180 |
. . . . . . . . . 10
class
0 |
12 | | c3 12345 |
. . . . . . . . . 10
class
3 |
13 | | cfzo 13707 |
. . . . . . . . . 10
class
..^ |
14 | 11, 12, 13 | co 7445 |
. . . . . . . . 9
class
(0..^3) |
15 | | vt |
. . . . . . . . . 10
setvar 𝑡 |
16 | 15 | cv 1536 |
. . . . . . . . 9
class 𝑡 |
17 | | vf |
. . . . . . . . . 10
setvar 𝑓 |
18 | 17 | cv 1536 |
. . . . . . . . 9
class 𝑓 |
19 | 14, 16, 18 | wf1o 6571 |
. . . . . . . 8
wff 𝑓:(0..^3)–1-1-onto→𝑡 |
20 | 11, 18 | cfv 6572 |
. . . . . . . . . . 11
class (𝑓‘0) |
21 | | c1 11181 |
. . . . . . . . . . . 12
class
1 |
22 | 21, 18 | cfv 6572 |
. . . . . . . . . . 11
class (𝑓‘1) |
23 | 20, 22 | cpr 4650 |
. . . . . . . . . 10
class {(𝑓‘0), (𝑓‘1)} |
24 | 8 | cv 1536 |
. . . . . . . . . 10
class 𝑒 |
25 | 23, 24 | wcel 2103 |
. . . . . . . . 9
wff {(𝑓‘0), (𝑓‘1)} ∈ 𝑒 |
26 | | c2 12344 |
. . . . . . . . . . . 12
class
2 |
27 | 26, 18 | cfv 6572 |
. . . . . . . . . . 11
class (𝑓‘2) |
28 | 20, 27 | cpr 4650 |
. . . . . . . . . 10
class {(𝑓‘0), (𝑓‘2)} |
29 | 28, 24 | wcel 2103 |
. . . . . . . . 9
wff {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 |
30 | 22, 27 | cpr 4650 |
. . . . . . . . . 10
class {(𝑓‘1), (𝑓‘2)} |
31 | 30, 24 | wcel 2103 |
. . . . . . . . 9
wff {(𝑓‘1), (𝑓‘2)} ∈ 𝑒 |
32 | 25, 29, 31 | w3a 1087 |
. . . . . . . 8
wff ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒) |
33 | 19, 32 | wa 395 |
. . . . . . 7
wff (𝑓:(0..^3)–1-1-onto→𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒)) |
34 | 33, 17 | wex 1777 |
. . . . . 6
wff
∃𝑓(𝑓:(0..^3)–1-1-onto→𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒)) |
35 | 4 | cv 1536 |
. . . . . . 7
class 𝑣 |
36 | 35 | cpw 4622 |
. . . . . 6
class 𝒫
𝑣 |
37 | 34, 15, 36 | crab 3438 |
. . . . 5
class {𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))} |
38 | 8, 10, 37 | csb 3915 |
. . . 4
class
⦋(Edg‘𝑔) / 𝑒⦌{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))} |
39 | 4, 7, 38 | csb 3915 |
. . 3
class
⦋(Vtx‘𝑔) / 𝑣⦌⦋(Edg‘𝑔) / 𝑒⦌{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))} |
40 | 2, 3, 39 | cmpt 5252 |
. 2
class (𝑔 ∈ V ↦
⦋(Vtx‘𝑔) / 𝑣⦌⦋(Edg‘𝑔) / 𝑒⦌{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))}) |
41 | 1, 40 | wceq 1537 |
1
wff GrTriangles
= (𝑔 ∈ V ↦
⦋(Vtx‘𝑔) / 𝑣⦌⦋(Edg‘𝑔) / 𝑒⦌{𝑡 ∈ 𝒫 𝑣 ∣ ∃𝑓(𝑓:(0..^3)–1-1-onto→𝑡 ∧ ({(𝑓‘0), (𝑓‘1)} ∈ 𝑒 ∧ {(𝑓‘0), (𝑓‘2)} ∈ 𝑒 ∧ {(𝑓‘1), (𝑓‘2)} ∈ 𝑒))}) |