Step | Hyp | Ref
| Expression |
1 | | usgrav 25867 |
. . 3
⊢ ({𝑉} USGrph 𝐸 → ({𝑉} ∈ V ∧ 𝐸 ∈ V)) |
2 | | simplr 788 |
. . . . 5
⊢
(((({𝑉} ∈ V
∧ 𝐸 ∈ V) ∧
{𝑉} USGrph 𝐸) ∧ 𝑉 ∈ 𝑋) → {𝑉} USGrph 𝐸) |
3 | | ral0 4028 |
. . . . . 6
⊢
∀𝑙 ∈
∅ ∃!𝑥 ∈
{𝑉} {{𝑥, 𝑉}, {𝑥, 𝑙}} ⊆ ran 𝐸 |
4 | | sneq 4135 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑉 → {𝑘} = {𝑉}) |
5 | 4 | difeq2d 3690 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑉 → ({𝑉} ∖ {𝑘}) = ({𝑉} ∖ {𝑉})) |
6 | | difid 3902 |
. . . . . . . . . 10
⊢ ({𝑉} ∖ {𝑉}) = ∅ |
7 | 5, 6 | syl6eq 2660 |
. . . . . . . . 9
⊢ (𝑘 = 𝑉 → ({𝑉} ∖ {𝑘}) = ∅) |
8 | | preq2 4213 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑉 → {𝑥, 𝑘} = {𝑥, 𝑉}) |
9 | 8 | preq1d 4218 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑉 → {{𝑥, 𝑘}, {𝑥, 𝑙}} = {{𝑥, 𝑉}, {𝑥, 𝑙}}) |
10 | 9 | sseq1d 3595 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑉 → ({{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ {{𝑥, 𝑉}, {𝑥, 𝑙}} ⊆ ran 𝐸)) |
11 | 10 | reubidv 3103 |
. . . . . . . . 9
⊢ (𝑘 = 𝑉 → (∃!𝑥 ∈ {𝑉} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∃!𝑥 ∈ {𝑉} {{𝑥, 𝑉}, {𝑥, 𝑙}} ⊆ ran 𝐸)) |
12 | 7, 11 | raleqbidv 3129 |
. . . . . . . 8
⊢ (𝑘 = 𝑉 → (∀𝑙 ∈ ({𝑉} ∖ {𝑘})∃!𝑥 ∈ {𝑉} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∀𝑙 ∈ ∅ ∃!𝑥 ∈ {𝑉} {{𝑥, 𝑉}, {𝑥, 𝑙}} ⊆ ran 𝐸)) |
13 | 12 | ralsng 4165 |
. . . . . . 7
⊢ (𝑉 ∈ 𝑋 → (∀𝑘 ∈ {𝑉}∀𝑙 ∈ ({𝑉} ∖ {𝑘})∃!𝑥 ∈ {𝑉} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∀𝑙 ∈ ∅ ∃!𝑥 ∈ {𝑉} {{𝑥, 𝑉}, {𝑥, 𝑙}} ⊆ ran 𝐸)) |
14 | 13 | adantl 481 |
. . . . . 6
⊢
(((({𝑉} ∈ V
∧ 𝐸 ∈ V) ∧
{𝑉} USGrph 𝐸) ∧ 𝑉 ∈ 𝑋) → (∀𝑘 ∈ {𝑉}∀𝑙 ∈ ({𝑉} ∖ {𝑘})∃!𝑥 ∈ {𝑉} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸 ↔ ∀𝑙 ∈ ∅ ∃!𝑥 ∈ {𝑉} {{𝑥, 𝑉}, {𝑥, 𝑙}} ⊆ ran 𝐸)) |
15 | 3, 14 | mpbiri 247 |
. . . . 5
⊢
(((({𝑉} ∈ V
∧ 𝐸 ∈ V) ∧
{𝑉} USGrph 𝐸) ∧ 𝑉 ∈ 𝑋) → ∀𝑘 ∈ {𝑉}∀𝑙 ∈ ({𝑉} ∖ {𝑘})∃!𝑥 ∈ {𝑉} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸) |
16 | | isfrgra 26517 |
. . . . . 6
⊢ (({𝑉} ∈ V ∧ 𝐸 ∈ V) → ({𝑉} FriendGrph 𝐸 ↔ ({𝑉} USGrph 𝐸 ∧ ∀𝑘 ∈ {𝑉}∀𝑙 ∈ ({𝑉} ∖ {𝑘})∃!𝑥 ∈ {𝑉} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸))) |
17 | 16 | ad2antrr 758 |
. . . . 5
⊢
(((({𝑉} ∈ V
∧ 𝐸 ∈ V) ∧
{𝑉} USGrph 𝐸) ∧ 𝑉 ∈ 𝑋) → ({𝑉} FriendGrph 𝐸 ↔ ({𝑉} USGrph 𝐸 ∧ ∀𝑘 ∈ {𝑉}∀𝑙 ∈ ({𝑉} ∖ {𝑘})∃!𝑥 ∈ {𝑉} {{𝑥, 𝑘}, {𝑥, 𝑙}} ⊆ ran 𝐸))) |
18 | 2, 15, 17 | mpbir2and 959 |
. . . 4
⊢
(((({𝑉} ∈ V
∧ 𝐸 ∈ V) ∧
{𝑉} USGrph 𝐸) ∧ 𝑉 ∈ 𝑋) → {𝑉} FriendGrph 𝐸) |
19 | 18 | ex 449 |
. . 3
⊢ ((({𝑉} ∈ V ∧ 𝐸 ∈ V) ∧ {𝑉} USGrph 𝐸) → (𝑉 ∈ 𝑋 → {𝑉} FriendGrph 𝐸)) |
20 | 1, 19 | mpancom 700 |
. 2
⊢ ({𝑉} USGrph 𝐸 → (𝑉 ∈ 𝑋 → {𝑉} FriendGrph 𝐸)) |
21 | 20 | impcom 445 |
1
⊢ ((𝑉 ∈ 𝑋 ∧ {𝑉} USGrph 𝐸) → {𝑉} FriendGrph 𝐸) |