Step |
Hyp |
Ref |
Expression |
1 |
|
umgrislfupgr.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
umgrislfupgr.i |
⊢ 𝐼 = ( iEdg ‘ 𝐺 ) |
3 |
|
umgrupgr |
⊢ ( 𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph ) |
4 |
1 2
|
umgrf |
⊢ ( 𝐺 ∈ UMGraph → 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } ) |
5 |
|
id |
⊢ ( 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } → 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } ) |
6 |
|
2re |
⊢ 2 ∈ ℝ |
7 |
6
|
leidi |
⊢ 2 ≤ 2 |
8 |
7
|
a1i |
⊢ ( ( ♯ ‘ 𝑥 ) = 2 → 2 ≤ 2 ) |
9 |
|
breq2 |
⊢ ( ( ♯ ‘ 𝑥 ) = 2 → ( 2 ≤ ( ♯ ‘ 𝑥 ) ↔ 2 ≤ 2 ) ) |
10 |
8 9
|
mpbird |
⊢ ( ( ♯ ‘ 𝑥 ) = 2 → 2 ≤ ( ♯ ‘ 𝑥 ) ) |
11 |
10
|
a1i |
⊢ ( 𝑥 ∈ 𝒫 𝑉 → ( ( ♯ ‘ 𝑥 ) = 2 → 2 ≤ ( ♯ ‘ 𝑥 ) ) ) |
12 |
11
|
ss2rabi |
⊢ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } ⊆ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } |
13 |
12
|
a1i |
⊢ ( 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } → { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } ⊆ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) |
14 |
5 13
|
fssd |
⊢ ( 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } → 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) |
15 |
4 14
|
syl |
⊢ ( 𝐺 ∈ UMGraph → 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) |
16 |
3 15
|
jca |
⊢ ( 𝐺 ∈ UMGraph → ( 𝐺 ∈ UPGraph ∧ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) ) |
17 |
1 2
|
upgrf |
⊢ ( 𝐺 ∈ UPGraph → 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ) |
18 |
|
fin |
⊢ ( 𝐼 : dom 𝐼 ⟶ ( { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ∩ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) ↔ ( 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ∧ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) ) |
19 |
|
umgrislfupgrlem |
⊢ ( { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ∩ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) = { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } |
20 |
|
feq3 |
⊢ ( ( { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ∩ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) = { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } → ( 𝐼 : dom 𝐼 ⟶ ( { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ∩ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) ↔ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ) ) |
21 |
19 20
|
ax-mp |
⊢ ( 𝐼 : dom 𝐼 ⟶ ( { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ∩ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) ↔ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ) |
22 |
18 21
|
sylbb1 |
⊢ ( ( 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } ∧ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) → 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ) |
23 |
17 22
|
sylan |
⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) → 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ) |
24 |
1 2
|
isumgr |
⊢ ( 𝐺 ∈ UPGraph → ( 𝐺 ∈ UMGraph ↔ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ) ) |
25 |
24
|
adantr |
⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) → ( 𝐺 ∈ UMGraph ↔ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ) ) |
26 |
23 25
|
mpbird |
⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) → 𝐺 ∈ UMGraph ) |
27 |
16 26
|
impbii |
⊢ ( 𝐺 ∈ UMGraph ↔ ( 𝐺 ∈ UPGraph ∧ 𝐼 : dom 𝐼 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ ( ♯ ‘ 𝑥 ) } ) ) |