Step |
Hyp |
Ref |
Expression |
1 |
|
edgval |
⊢ ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 ) |
2 |
1
|
a1i |
⊢ ( 𝐺 ∈ UPGraph → ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 ) ) |
3 |
2
|
eleq2d |
⊢ ( 𝐺 ∈ UPGraph → ( 𝐸 ∈ ( Edg ‘ 𝐺 ) ↔ 𝐸 ∈ ran ( iEdg ‘ 𝐺 ) ) ) |
4 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
5 |
|
eqid |
⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 ) |
6 |
4 5
|
upgrf |
⊢ ( 𝐺 ∈ UPGraph → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) |
7 |
6
|
frnd |
⊢ ( 𝐺 ∈ UPGraph → ran ( iEdg ‘ 𝐺 ) ⊆ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) |
8 |
7
|
sseld |
⊢ ( 𝐺 ∈ UPGraph → ( 𝐸 ∈ ran ( iEdg ‘ 𝐺 ) → 𝐸 ∈ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) ) |
9 |
|
fveq2 |
⊢ ( 𝑥 = 𝐸 → ( ♯ ‘ 𝑥 ) = ( ♯ ‘ 𝐸 ) ) |
10 |
9
|
breq1d |
⊢ ( 𝑥 = 𝐸 → ( ( ♯ ‘ 𝑥 ) ≤ 2 ↔ ( ♯ ‘ 𝐸 ) ≤ 2 ) ) |
11 |
10
|
elrab |
⊢ ( 𝐸 ∈ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ↔ ( 𝐸 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∧ ( ♯ ‘ 𝐸 ) ≤ 2 ) ) |
12 |
|
eldifsn |
⊢ ( 𝐸 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ↔ ( 𝐸 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∧ 𝐸 ≠ ∅ ) ) |
13 |
12
|
biimpi |
⊢ ( 𝐸 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) → ( 𝐸 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∧ 𝐸 ≠ ∅ ) ) |
14 |
13
|
anim1i |
⊢ ( ( 𝐸 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∧ ( ♯ ‘ 𝐸 ) ≤ 2 ) → ( ( 𝐸 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∧ 𝐸 ≠ ∅ ) ∧ ( ♯ ‘ 𝐸 ) ≤ 2 ) ) |
15 |
|
df-3an |
⊢ ( ( 𝐸 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∧ 𝐸 ≠ ∅ ∧ ( ♯ ‘ 𝐸 ) ≤ 2 ) ↔ ( ( 𝐸 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∧ 𝐸 ≠ ∅ ) ∧ ( ♯ ‘ 𝐸 ) ≤ 2 ) ) |
16 |
14 15
|
sylibr |
⊢ ( ( 𝐸 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∧ ( ♯ ‘ 𝐸 ) ≤ 2 ) → ( 𝐸 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∧ 𝐸 ≠ ∅ ∧ ( ♯ ‘ 𝐸 ) ≤ 2 ) ) |
17 |
16
|
a1i |
⊢ ( 𝐺 ∈ UPGraph → ( ( 𝐸 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∧ ( ♯ ‘ 𝐸 ) ≤ 2 ) → ( 𝐸 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∧ 𝐸 ≠ ∅ ∧ ( ♯ ‘ 𝐸 ) ≤ 2 ) ) ) |
18 |
11 17
|
syl5bi |
⊢ ( 𝐺 ∈ UPGraph → ( 𝐸 ∈ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } → ( 𝐸 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∧ 𝐸 ≠ ∅ ∧ ( ♯ ‘ 𝐸 ) ≤ 2 ) ) ) |
19 |
8 18
|
syld |
⊢ ( 𝐺 ∈ UPGraph → ( 𝐸 ∈ ran ( iEdg ‘ 𝐺 ) → ( 𝐸 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∧ 𝐸 ≠ ∅ ∧ ( ♯ ‘ 𝐸 ) ≤ 2 ) ) ) |
20 |
3 19
|
sylbid |
⊢ ( 𝐺 ∈ UPGraph → ( 𝐸 ∈ ( Edg ‘ 𝐺 ) → ( 𝐸 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∧ 𝐸 ≠ ∅ ∧ ( ♯ ‘ 𝐸 ) ≤ 2 ) ) ) |
21 |
20
|
imp |
⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐸 ∈ ( Edg ‘ 𝐺 ) ) → ( 𝐸 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∧ 𝐸 ≠ ∅ ∧ ( ♯ ‘ 𝐸 ) ≤ 2 ) ) |