Step |
Hyp |
Ref |
Expression |
1 |
|
structvtxvallem.s |
⊢ 𝑆 ∈ ℕ |
2 |
|
structvtxvallem.b |
⊢ ( Base ‘ ndx ) < 𝑆 |
3 |
|
structvtxvallem.g |
⊢ 𝐺 = { 〈 ( Base ‘ ndx ) , 𝑉 〉 , 〈 𝑆 , 𝐸 〉 } |
4 |
3 2 1
|
2strstr1 |
⊢ 𝐺 Struct 〈 ( Base ‘ ndx ) , 𝑆 〉 |
5 |
|
structn0fun |
⊢ ( 𝐺 Struct 〈 ( Base ‘ ndx ) , 𝑆 〉 → Fun ( 𝐺 ∖ { ∅ } ) ) |
6 |
1 2 3
|
structvtxvallem |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ) → 2 ≤ ( ♯ ‘ dom 𝐺 ) ) |
7 |
|
funiedgdmge2val |
⊢ ( ( Fun ( 𝐺 ∖ { ∅ } ) ∧ 2 ≤ ( ♯ ‘ dom 𝐺 ) ) → ( iEdg ‘ 𝐺 ) = ( .ef ‘ 𝐺 ) ) |
8 |
5 6 7
|
syl2an |
⊢ ( ( 𝐺 Struct 〈 ( Base ‘ ndx ) , 𝑆 〉 ∧ ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ) ) → ( iEdg ‘ 𝐺 ) = ( .ef ‘ 𝐺 ) ) |
9 |
4 8
|
mpan |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ) → ( iEdg ‘ 𝐺 ) = ( .ef ‘ 𝐺 ) ) |
10 |
9
|
3adant3 |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ ( .ef ‘ ndx ) ) → ( iEdg ‘ 𝐺 ) = ( .ef ‘ 𝐺 ) ) |
11 |
|
prex |
⊢ { 〈 ( Base ‘ ndx ) , 𝑉 〉 , 〈 𝑆 , 𝐸 〉 } ∈ V |
12 |
11
|
a1i |
⊢ ( 𝐺 = { 〈 ( Base ‘ ndx ) , 𝑉 〉 , 〈 𝑆 , 𝐸 〉 } → { 〈 ( Base ‘ ndx ) , 𝑉 〉 , 〈 𝑆 , 𝐸 〉 } ∈ V ) |
13 |
3 12
|
eqeltrid |
⊢ ( 𝐺 = { 〈 ( Base ‘ ndx ) , 𝑉 〉 , 〈 𝑆 , 𝐸 〉 } → 𝐺 ∈ V ) |
14 |
|
edgfndxid |
⊢ ( 𝐺 ∈ V → ( .ef ‘ 𝐺 ) = ( 𝐺 ‘ ( .ef ‘ ndx ) ) ) |
15 |
3 13 14
|
mp2b |
⊢ ( .ef ‘ 𝐺 ) = ( 𝐺 ‘ ( .ef ‘ ndx ) ) |
16 |
|
basendxnedgfndx |
⊢ ( Base ‘ ndx ) ≠ ( .ef ‘ ndx ) |
17 |
16
|
nesymi |
⊢ ¬ ( .ef ‘ ndx ) = ( Base ‘ ndx ) |
18 |
17
|
a1i |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ ( .ef ‘ ndx ) ) → ¬ ( .ef ‘ ndx ) = ( Base ‘ ndx ) ) |
19 |
|
neneq |
⊢ ( 𝑆 ≠ ( .ef ‘ ndx ) → ¬ 𝑆 = ( .ef ‘ ndx ) ) |
20 |
|
eqcom |
⊢ ( ( .ef ‘ ndx ) = 𝑆 ↔ 𝑆 = ( .ef ‘ ndx ) ) |
21 |
19 20
|
sylnibr |
⊢ ( 𝑆 ≠ ( .ef ‘ ndx ) → ¬ ( .ef ‘ ndx ) = 𝑆 ) |
22 |
21
|
3ad2ant3 |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ ( .ef ‘ ndx ) ) → ¬ ( .ef ‘ ndx ) = 𝑆 ) |
23 |
|
ioran |
⊢ ( ¬ ( ( .ef ‘ ndx ) = ( Base ‘ ndx ) ∨ ( .ef ‘ ndx ) = 𝑆 ) ↔ ( ¬ ( .ef ‘ ndx ) = ( Base ‘ ndx ) ∧ ¬ ( .ef ‘ ndx ) = 𝑆 ) ) |
24 |
18 22 23
|
sylanbrc |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ ( .ef ‘ ndx ) ) → ¬ ( ( .ef ‘ ndx ) = ( Base ‘ ndx ) ∨ ( .ef ‘ ndx ) = 𝑆 ) ) |
25 |
|
fvex |
⊢ ( .ef ‘ ndx ) ∈ V |
26 |
25
|
elpr |
⊢ ( ( .ef ‘ ndx ) ∈ { ( Base ‘ ndx ) , 𝑆 } ↔ ( ( .ef ‘ ndx ) = ( Base ‘ ndx ) ∨ ( .ef ‘ ndx ) = 𝑆 ) ) |
27 |
24 26
|
sylnibr |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ ( .ef ‘ ndx ) ) → ¬ ( .ef ‘ ndx ) ∈ { ( Base ‘ ndx ) , 𝑆 } ) |
28 |
3
|
dmeqi |
⊢ dom 𝐺 = dom { 〈 ( Base ‘ ndx ) , 𝑉 〉 , 〈 𝑆 , 𝐸 〉 } |
29 |
|
dmpropg |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ) → dom { 〈 ( Base ‘ ndx ) , 𝑉 〉 , 〈 𝑆 , 𝐸 〉 } = { ( Base ‘ ndx ) , 𝑆 } ) |
30 |
29
|
3adant3 |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ ( .ef ‘ ndx ) ) → dom { 〈 ( Base ‘ ndx ) , 𝑉 〉 , 〈 𝑆 , 𝐸 〉 } = { ( Base ‘ ndx ) , 𝑆 } ) |
31 |
28 30
|
eqtrid |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ ( .ef ‘ ndx ) ) → dom 𝐺 = { ( Base ‘ ndx ) , 𝑆 } ) |
32 |
27 31
|
neleqtrrd |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ ( .ef ‘ ndx ) ) → ¬ ( .ef ‘ ndx ) ∈ dom 𝐺 ) |
33 |
|
ndmfv |
⊢ ( ¬ ( .ef ‘ ndx ) ∈ dom 𝐺 → ( 𝐺 ‘ ( .ef ‘ ndx ) ) = ∅ ) |
34 |
32 33
|
syl |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ ( .ef ‘ ndx ) ) → ( 𝐺 ‘ ( .ef ‘ ndx ) ) = ∅ ) |
35 |
15 34
|
eqtrid |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ ( .ef ‘ ndx ) ) → ( .ef ‘ 𝐺 ) = ∅ ) |
36 |
10 35
|
eqtrd |
⊢ ( ( 𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ ( .ef ‘ ndx ) ) → ( iEdg ‘ 𝐺 ) = ∅ ) |