Step |
Hyp |
Ref |
Expression |
1 |
|
isumgr.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
isumgr.e |
⊢ 𝐸 = ( iEdg ‘ 𝐺 ) |
3 |
1 2
|
isumgrs |
⊢ ( 𝐺 ∈ 𝑈 → ( 𝐺 ∈ UMGraph ↔ 𝐸 : dom 𝐸 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } ) ) |
4 |
3
|
adantr |
⊢ ( ( 𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋 ) → ( 𝐺 ∈ UMGraph ↔ 𝐸 : dom 𝐸 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } ) ) |
5 |
|
wrdf |
⊢ ( 𝐸 ∈ Word 𝑋 → 𝐸 : ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ⟶ 𝑋 ) |
6 |
5
|
adantl |
⊢ ( ( 𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋 ) → 𝐸 : ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ⟶ 𝑋 ) |
7 |
6
|
fdmd |
⊢ ( ( 𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋 ) → dom 𝐸 = ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ) |
8 |
7
|
feq2d |
⊢ ( ( 𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋 ) → ( 𝐸 : dom 𝐸 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } ↔ 𝐸 : ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } ) ) |
9 |
|
iswrdi |
⊢ ( 𝐸 : ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } → 𝐸 ∈ Word { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } ) |
10 |
|
wrdf |
⊢ ( 𝐸 ∈ Word { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } → 𝐸 : ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } ) |
11 |
9 10
|
impbii |
⊢ ( 𝐸 : ( 0 ..^ ( ♯ ‘ 𝐸 ) ) ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } ↔ 𝐸 ∈ Word { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } ) |
12 |
8 11
|
bitrdi |
⊢ ( ( 𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋 ) → ( 𝐸 : dom 𝐸 ⟶ { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } ↔ 𝐸 ∈ Word { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } ) ) |
13 |
4 12
|
bitrd |
⊢ ( ( 𝐺 ∈ 𝑈 ∧ 𝐸 ∈ Word 𝑋 ) → ( 𝐺 ∈ UMGraph ↔ 𝐸 ∈ Word { 𝑥 ∈ 𝒫 𝑉 ∣ ( ♯ ‘ 𝑥 ) = 2 } ) ) |