| Step |
Hyp |
Ref |
Expression |
| 1 |
|
grtri.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
| 2 |
|
grtri.e |
⊢ 𝐸 = ( Edg ‘ 𝐺 ) |
| 3 |
1 2
|
grtriprop |
⊢ ( 𝑇 ∈ ( GrTriangles ‘ 𝐺 ) → ∃ 𝑥 ∈ 𝑉 ∃ 𝑦 ∈ 𝑉 ∃ 𝑧 ∈ 𝑉 ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } ∧ ( ♯ ‘ 𝑇 ) = 3 ∧ ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 𝐸 ) ) ) |
| 4 |
|
df-tp |
⊢ { 𝑥 , 𝑦 , 𝑧 } = ( { 𝑥 , 𝑦 } ∪ { 𝑧 } ) |
| 5 |
|
prelpwi |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) → { 𝑥 , 𝑦 } ∈ 𝒫 𝑉 ) |
| 6 |
|
snelpwi |
⊢ ( 𝑧 ∈ 𝑉 → { 𝑧 } ∈ 𝒫 𝑉 ) |
| 7 |
5 6
|
anim12i |
⊢ ( ( ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ) ∧ 𝑧 ∈ 𝑉 ) → ( { 𝑥 , 𝑦 } ∈ 𝒫 𝑉 ∧ { 𝑧 } ∈ 𝒫 𝑉 ) ) |
| 8 |
7
|
anasss |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( { 𝑥 , 𝑦 } ∈ 𝒫 𝑉 ∧ { 𝑧 } ∈ 𝒫 𝑉 ) ) |
| 9 |
|
pwuncl |
⊢ ( ( { 𝑥 , 𝑦 } ∈ 𝒫 𝑉 ∧ { 𝑧 } ∈ 𝒫 𝑉 ) → ( { 𝑥 , 𝑦 } ∪ { 𝑧 } ) ∈ 𝒫 𝑉 ) |
| 10 |
8 9
|
syl |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( { 𝑥 , 𝑦 } ∪ { 𝑧 } ) ∈ 𝒫 𝑉 ) |
| 11 |
4 10
|
eqeltrid |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → { 𝑥 , 𝑦 , 𝑧 } ∈ 𝒫 𝑉 ) |
| 12 |
11
|
adantr |
⊢ ( ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } ∧ ( ♯ ‘ 𝑇 ) = 3 ∧ ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 𝐸 ) ) ) → { 𝑥 , 𝑦 , 𝑧 } ∈ 𝒫 𝑉 ) |
| 13 |
|
eleq1 |
⊢ ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } → ( 𝑇 ∈ 𝒫 𝑉 ↔ { 𝑥 , 𝑦 , 𝑧 } ∈ 𝒫 𝑉 ) ) |
| 14 |
13
|
3ad2ant1 |
⊢ ( ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } ∧ ( ♯ ‘ 𝑇 ) = 3 ∧ ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 𝐸 ) ) → ( 𝑇 ∈ 𝒫 𝑉 ↔ { 𝑥 , 𝑦 , 𝑧 } ∈ 𝒫 𝑉 ) ) |
| 15 |
14
|
adantl |
⊢ ( ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } ∧ ( ♯ ‘ 𝑇 ) = 3 ∧ ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 𝐸 ) ) ) → ( 𝑇 ∈ 𝒫 𝑉 ↔ { 𝑥 , 𝑦 , 𝑧 } ∈ 𝒫 𝑉 ) ) |
| 16 |
12 15
|
mpbird |
⊢ ( ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } ∧ ( ♯ ‘ 𝑇 ) = 3 ∧ ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 𝐸 ) ) ) → 𝑇 ∈ 𝒫 𝑉 ) |
| 17 |
|
ovex |
⊢ ( 0 ..^ 3 ) ∈ V |
| 18 |
17
|
mptex |
⊢ ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ∈ V |
| 19 |
18
|
a1i |
⊢ ( ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } ∧ ( ♯ ‘ 𝑇 ) = 3 ∧ ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 𝐸 ) ) ) → ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ∈ V ) |
| 20 |
|
3anass |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ↔ ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ) |
| 21 |
20
|
biimpri |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) |
| 22 |
|
fveq2 |
⊢ ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } → ( ♯ ‘ 𝑇 ) = ( ♯ ‘ { 𝑥 , 𝑦 , 𝑧 } ) ) |
| 23 |
22
|
eqcomd |
⊢ ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } → ( ♯ ‘ { 𝑥 , 𝑦 , 𝑧 } ) = ( ♯ ‘ 𝑇 ) ) |
| 24 |
23
|
3ad2ant1 |
⊢ ( ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } ∧ ( ♯ ‘ 𝑇 ) = 3 ∧ ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 𝐸 ) ) → ( ♯ ‘ { 𝑥 , 𝑦 , 𝑧 } ) = ( ♯ ‘ 𝑇 ) ) |
| 25 |
|
simp2 |
⊢ ( ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } ∧ ( ♯ ‘ 𝑇 ) = 3 ∧ ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 𝐸 ) ) → ( ♯ ‘ 𝑇 ) = 3 ) |
| 26 |
24 25
|
eqtrd |
⊢ ( ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } ∧ ( ♯ ‘ 𝑇 ) = 3 ∧ ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 𝐸 ) ) → ( ♯ ‘ { 𝑥 , 𝑦 , 𝑧 } ) = 3 ) |
| 27 |
|
eqid |
⊢ ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) = ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) |
| 28 |
|
eqid |
⊢ { 𝑥 , 𝑦 , 𝑧 } = { 𝑥 , 𝑦 , 𝑧 } |
| 29 |
27 28
|
tpf1o |
⊢ ( ( ( 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ∧ ( ♯ ‘ { 𝑥 , 𝑦 , 𝑧 } ) = 3 ) → ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) : ( 0 ..^ 3 ) –1-1-onto→ { 𝑥 , 𝑦 , 𝑧 } ) |
| 30 |
21 26 29
|
syl2an |
⊢ ( ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } ∧ ( ♯ ‘ 𝑇 ) = 3 ∧ ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 𝐸 ) ) ) → ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) : ( 0 ..^ 3 ) –1-1-onto→ { 𝑥 , 𝑦 , 𝑧 } ) |
| 31 |
|
f1oeq3 |
⊢ ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } → ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) : ( 0 ..^ 3 ) –1-1-onto→ 𝑇 ↔ ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) : ( 0 ..^ 3 ) –1-1-onto→ { 𝑥 , 𝑦 , 𝑧 } ) ) |
| 32 |
31
|
3ad2ant1 |
⊢ ( ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } ∧ ( ♯ ‘ 𝑇 ) = 3 ∧ ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 𝐸 ) ) → ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) : ( 0 ..^ 3 ) –1-1-onto→ 𝑇 ↔ ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) : ( 0 ..^ 3 ) –1-1-onto→ { 𝑥 , 𝑦 , 𝑧 } ) ) |
| 33 |
32
|
adantl |
⊢ ( ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } ∧ ( ♯ ‘ 𝑇 ) = 3 ∧ ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 𝐸 ) ) ) → ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) : ( 0 ..^ 3 ) –1-1-onto→ 𝑇 ↔ ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) : ( 0 ..^ 3 ) –1-1-onto→ { 𝑥 , 𝑦 , 𝑧 } ) ) |
| 34 |
30 33
|
mpbird |
⊢ ( ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } ∧ ( ♯ ‘ 𝑇 ) = 3 ∧ ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 𝐸 ) ) ) → ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) : ( 0 ..^ 3 ) –1-1-onto→ 𝑇 ) |
| 35 |
27
|
tpf1ofv0 |
⊢ ( 𝑥 ∈ 𝑉 → ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) = 𝑥 ) |
| 36 |
35
|
adantr |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) = 𝑥 ) |
| 37 |
27
|
tpf1ofv1 |
⊢ ( 𝑦 ∈ 𝑉 → ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) = 𝑦 ) |
| 38 |
37
|
adantr |
⊢ ( ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) → ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) = 𝑦 ) |
| 39 |
38
|
adantl |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) = 𝑦 ) |
| 40 |
36 39
|
preq12d |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) } = { 𝑥 , 𝑦 } ) |
| 41 |
40
|
eqcomd |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → { 𝑥 , 𝑦 } = { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) } ) |
| 42 |
41
|
eleq1d |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( { 𝑥 , 𝑦 } ∈ 𝐸 ↔ { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) } ∈ 𝐸 ) ) |
| 43 |
27
|
tpf1ofv2 |
⊢ ( 𝑧 ∈ 𝑉 → ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) = 𝑧 ) |
| 44 |
43
|
adantl |
⊢ ( ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) → ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) = 𝑧 ) |
| 45 |
44
|
adantl |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) = 𝑧 ) |
| 46 |
36 45
|
preq12d |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) } = { 𝑥 , 𝑧 } ) |
| 47 |
46
|
eqcomd |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → { 𝑥 , 𝑧 } = { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) } ) |
| 48 |
47
|
eleq1d |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( { 𝑥 , 𝑧 } ∈ 𝐸 ↔ { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) } ∈ 𝐸 ) ) |
| 49 |
39 45
|
preq12d |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) } = { 𝑦 , 𝑧 } ) |
| 50 |
49
|
eqcomd |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → { 𝑦 , 𝑧 } = { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) } ) |
| 51 |
50
|
eleq1d |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( { 𝑦 , 𝑧 } ∈ 𝐸 ↔ { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) } ∈ 𝐸 ) ) |
| 52 |
42 48 51
|
3anbi123d |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 𝐸 ) ↔ ( { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) } ∈ 𝐸 ∧ { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) } ∈ 𝐸 ∧ { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) } ∈ 𝐸 ) ) ) |
| 53 |
52
|
biimpd |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 𝐸 ) → ( { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) } ∈ 𝐸 ∧ { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) } ∈ 𝐸 ∧ { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) } ∈ 𝐸 ) ) ) |
| 54 |
53
|
2a1d |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } → ( ( ♯ ‘ 𝑇 ) = 3 → ( ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 𝐸 ) → ( { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) } ∈ 𝐸 ∧ { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) } ∈ 𝐸 ∧ { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) } ∈ 𝐸 ) ) ) ) ) |
| 55 |
54
|
3imp2 |
⊢ ( ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } ∧ ( ♯ ‘ 𝑇 ) = 3 ∧ ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 𝐸 ) ) ) → ( { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) } ∈ 𝐸 ∧ { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) } ∈ 𝐸 ∧ { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) } ∈ 𝐸 ) ) |
| 56 |
34 55
|
jca |
⊢ ( ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } ∧ ( ♯ ‘ 𝑇 ) = 3 ∧ ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 𝐸 ) ) ) → ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) : ( 0 ..^ 3 ) –1-1-onto→ 𝑇 ∧ ( { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) } ∈ 𝐸 ∧ { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) } ∈ 𝐸 ∧ { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) } ∈ 𝐸 ) ) ) |
| 57 |
|
f1oeq1 |
⊢ ( 𝑓 = ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) → ( 𝑓 : ( 0 ..^ 3 ) –1-1-onto→ 𝑇 ↔ ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) : ( 0 ..^ 3 ) –1-1-onto→ 𝑇 ) ) |
| 58 |
|
fveq1 |
⊢ ( 𝑓 = ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) → ( 𝑓 ‘ 0 ) = ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) ) |
| 59 |
|
fveq1 |
⊢ ( 𝑓 = ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) → ( 𝑓 ‘ 1 ) = ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) ) |
| 60 |
58 59
|
preq12d |
⊢ ( 𝑓 = ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) → { ( 𝑓 ‘ 0 ) , ( 𝑓 ‘ 1 ) } = { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) } ) |
| 61 |
60
|
eleq1d |
⊢ ( 𝑓 = ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) → ( { ( 𝑓 ‘ 0 ) , ( 𝑓 ‘ 1 ) } ∈ 𝐸 ↔ { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) } ∈ 𝐸 ) ) |
| 62 |
|
fveq1 |
⊢ ( 𝑓 = ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) → ( 𝑓 ‘ 2 ) = ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) ) |
| 63 |
58 62
|
preq12d |
⊢ ( 𝑓 = ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) → { ( 𝑓 ‘ 0 ) , ( 𝑓 ‘ 2 ) } = { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) } ) |
| 64 |
63
|
eleq1d |
⊢ ( 𝑓 = ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) → ( { ( 𝑓 ‘ 0 ) , ( 𝑓 ‘ 2 ) } ∈ 𝐸 ↔ { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) } ∈ 𝐸 ) ) |
| 65 |
59 62
|
preq12d |
⊢ ( 𝑓 = ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) → { ( 𝑓 ‘ 1 ) , ( 𝑓 ‘ 2 ) } = { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) } ) |
| 66 |
65
|
eleq1d |
⊢ ( 𝑓 = ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) → ( { ( 𝑓 ‘ 1 ) , ( 𝑓 ‘ 2 ) } ∈ 𝐸 ↔ { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) } ∈ 𝐸 ) ) |
| 67 |
61 64 66
|
3anbi123d |
⊢ ( 𝑓 = ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) → ( ( { ( 𝑓 ‘ 0 ) , ( 𝑓 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑓 ‘ 0 ) , ( 𝑓 ‘ 2 ) } ∈ 𝐸 ∧ { ( 𝑓 ‘ 1 ) , ( 𝑓 ‘ 2 ) } ∈ 𝐸 ) ↔ ( { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) } ∈ 𝐸 ∧ { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) } ∈ 𝐸 ∧ { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) } ∈ 𝐸 ) ) ) |
| 68 |
57 67
|
anbi12d |
⊢ ( 𝑓 = ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) → ( ( 𝑓 : ( 0 ..^ 3 ) –1-1-onto→ 𝑇 ∧ ( { ( 𝑓 ‘ 0 ) , ( 𝑓 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑓 ‘ 0 ) , ( 𝑓 ‘ 2 ) } ∈ 𝐸 ∧ { ( 𝑓 ‘ 1 ) , ( 𝑓 ‘ 2 ) } ∈ 𝐸 ) ) ↔ ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) : ( 0 ..^ 3 ) –1-1-onto→ 𝑇 ∧ ( { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) } ∈ 𝐸 ∧ { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 0 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) } ∈ 𝐸 ∧ { ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 1 ) , ( ( 𝑖 ∈ ( 0 ..^ 3 ) ↦ if ( 𝑖 = 0 , 𝑥 , if ( 𝑖 = 1 , 𝑦 , 𝑧 ) ) ) ‘ 2 ) } ∈ 𝐸 ) ) ) ) |
| 69 |
19 56 68
|
spcedv |
⊢ ( ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } ∧ ( ♯ ‘ 𝑇 ) = 3 ∧ ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 𝐸 ) ) ) → ∃ 𝑓 ( 𝑓 : ( 0 ..^ 3 ) –1-1-onto→ 𝑇 ∧ ( { ( 𝑓 ‘ 0 ) , ( 𝑓 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑓 ‘ 0 ) , ( 𝑓 ‘ 2 ) } ∈ 𝐸 ∧ { ( 𝑓 ‘ 1 ) , ( 𝑓 ‘ 2 ) } ∈ 𝐸 ) ) ) |
| 70 |
16 69
|
jca |
⊢ ( ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } ∧ ( ♯ ‘ 𝑇 ) = 3 ∧ ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 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 ) ) ) ) |
| 71 |
1
|
1vgrex |
⊢ ( 𝑥 ∈ 𝑉 → 𝐺 ∈ V ) |
| 72 |
1 2
|
grtri |
⊢ ( 𝐺 ∈ V → ( GrTriangles ‘ 𝐺 ) = { 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 ) ) } ) |
| 73 |
72
|
eleq2d |
⊢ ( 𝐺 ∈ V → ( 𝑇 ∈ ( GrTriangles ‘ 𝐺 ) ↔ 𝑇 ∈ { 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 ) ) } ) ) |
| 74 |
71 73
|
syl |
⊢ ( 𝑥 ∈ 𝑉 → ( 𝑇 ∈ ( GrTriangles ‘ 𝐺 ) ↔ 𝑇 ∈ { 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 ) ) } ) ) |
| 75 |
|
f1oeq3 |
⊢ ( 𝑡 = 𝑇 → ( 𝑓 : ( 0 ..^ 3 ) –1-1-onto→ 𝑡 ↔ 𝑓 : ( 0 ..^ 3 ) –1-1-onto→ 𝑇 ) ) |
| 76 |
75
|
anbi1d |
⊢ ( 𝑡 = 𝑇 → ( ( 𝑓 : ( 0 ..^ 3 ) –1-1-onto→ 𝑡 ∧ ( { ( 𝑓 ‘ 0 ) , ( 𝑓 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑓 ‘ 0 ) , ( 𝑓 ‘ 2 ) } ∈ 𝐸 ∧ { ( 𝑓 ‘ 1 ) , ( 𝑓 ‘ 2 ) } ∈ 𝐸 ) ) ↔ ( 𝑓 : ( 0 ..^ 3 ) –1-1-onto→ 𝑇 ∧ ( { ( 𝑓 ‘ 0 ) , ( 𝑓 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑓 ‘ 0 ) , ( 𝑓 ‘ 2 ) } ∈ 𝐸 ∧ { ( 𝑓 ‘ 1 ) , ( 𝑓 ‘ 2 ) } ∈ 𝐸 ) ) ) ) |
| 77 |
76
|
exbidv |
⊢ ( 𝑡 = 𝑇 → ( ∃ 𝑓 ( 𝑓 : ( 0 ..^ 3 ) –1-1-onto→ 𝑡 ∧ ( { ( 𝑓 ‘ 0 ) , ( 𝑓 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑓 ‘ 0 ) , ( 𝑓 ‘ 2 ) } ∈ 𝐸 ∧ { ( 𝑓 ‘ 1 ) , ( 𝑓 ‘ 2 ) } ∈ 𝐸 ) ) ↔ ∃ 𝑓 ( 𝑓 : ( 0 ..^ 3 ) –1-1-onto→ 𝑇 ∧ ( { ( 𝑓 ‘ 0 ) , ( 𝑓 ‘ 1 ) } ∈ 𝐸 ∧ { ( 𝑓 ‘ 0 ) , ( 𝑓 ‘ 2 ) } ∈ 𝐸 ∧ { ( 𝑓 ‘ 1 ) , ( 𝑓 ‘ 2 ) } ∈ 𝐸 ) ) ) ) |
| 78 |
77
|
elrab |
⊢ ( 𝑇 ∈ { 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 ) ) } ↔ ( 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 ) ) ) ) |
| 79 |
74 78
|
bitrdi |
⊢ ( 𝑥 ∈ 𝑉 → ( 𝑇 ∈ ( GrTriangles ‘ 𝐺 ) ↔ ( 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 ) ) ) ) ) |
| 80 |
79
|
adantr |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( 𝑇 ∈ ( GrTriangles ‘ 𝐺 ) ↔ ( 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 ) ) ) ) ) |
| 81 |
80
|
adantr |
⊢ ( ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } ∧ ( ♯ ‘ 𝑇 ) = 3 ∧ ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 𝐸 ) ) ) → ( 𝑇 ∈ ( GrTriangles ‘ 𝐺 ) ↔ ( 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 ) ) ) ) ) |
| 82 |
70 81
|
mpbird |
⊢ ( ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) ∧ ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } ∧ ( ♯ ‘ 𝑇 ) = 3 ∧ ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 𝐸 ) ) ) → 𝑇 ∈ ( GrTriangles ‘ 𝐺 ) ) |
| 83 |
82
|
ex |
⊢ ( ( 𝑥 ∈ 𝑉 ∧ ( 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ) ) → ( ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } ∧ ( ♯ ‘ 𝑇 ) = 3 ∧ ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 𝐸 ) ) → 𝑇 ∈ ( GrTriangles ‘ 𝐺 ) ) ) |
| 84 |
83
|
rexlimdvva |
⊢ ( 𝑥 ∈ 𝑉 → ( ∃ 𝑦 ∈ 𝑉 ∃ 𝑧 ∈ 𝑉 ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } ∧ ( ♯ ‘ 𝑇 ) = 3 ∧ ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 𝐸 ) ) → 𝑇 ∈ ( GrTriangles ‘ 𝐺 ) ) ) |
| 85 |
84
|
rexlimiv |
⊢ ( ∃ 𝑥 ∈ 𝑉 ∃ 𝑦 ∈ 𝑉 ∃ 𝑧 ∈ 𝑉 ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } ∧ ( ♯ ‘ 𝑇 ) = 3 ∧ ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 𝐸 ) ) → 𝑇 ∈ ( GrTriangles ‘ 𝐺 ) ) |
| 86 |
3 85
|
impbii |
⊢ ( 𝑇 ∈ ( GrTriangles ‘ 𝐺 ) ↔ ∃ 𝑥 ∈ 𝑉 ∃ 𝑦 ∈ 𝑉 ∃ 𝑧 ∈ 𝑉 ( 𝑇 = { 𝑥 , 𝑦 , 𝑧 } ∧ ( ♯ ‘ 𝑇 ) = 3 ∧ ( { 𝑥 , 𝑦 } ∈ 𝐸 ∧ { 𝑥 , 𝑧 } ∈ 𝐸 ∧ { 𝑦 , 𝑧 } ∈ 𝐸 ) ) ) |