Step |
Hyp |
Ref |
Expression |
1 |
|
isupgr.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
isupgr.e |
⊢ 𝐸 = ( iEdg ‘ 𝐺 ) |
3 |
1 2
|
upgrfn |
⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ) → 𝐸 : 𝐴 ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) |
4 |
3
|
ffvelrnda |
⊢ ( ( ( 𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ) ∧ 𝐹 ∈ 𝐴 ) → ( 𝐸 ‘ 𝐹 ) ∈ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) |
5 |
4
|
3impa |
⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴 ) → ( 𝐸 ‘ 𝐹 ) ∈ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) |
6 |
|
fveq2 |
⊢ ( 𝑥 = ( 𝐸 ‘ 𝐹 ) → ( ♯ ‘ 𝑥 ) = ( ♯ ‘ ( 𝐸 ‘ 𝐹 ) ) ) |
7 |
6
|
breq1d |
⊢ ( 𝑥 = ( 𝐸 ‘ 𝐹 ) → ( ( ♯ ‘ 𝑥 ) ≤ 2 ↔ ( ♯ ‘ ( 𝐸 ‘ 𝐹 ) ) ≤ 2 ) ) |
8 |
7
|
elrab |
⊢ ( ( 𝐸 ‘ 𝐹 ) ∈ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ↔ ( ( 𝐸 ‘ 𝐹 ) ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∧ ( ♯ ‘ ( 𝐸 ‘ 𝐹 ) ) ≤ 2 ) ) |
9 |
8
|
simprbi |
⊢ ( ( 𝐸 ‘ 𝐹 ) ∈ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } → ( ♯ ‘ ( 𝐸 ‘ 𝐹 ) ) ≤ 2 ) |
10 |
5 9
|
syl |
⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴 ) → ( ♯ ‘ ( 𝐸 ‘ 𝐹 ) ) ≤ 2 ) |