| Step |
Hyp |
Ref |
Expression |
| 1 |
|
isubgrvtx.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
| 2 |
|
eqid |
⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 ) |
| 3 |
1 2
|
uhgrf |
⊢ ( 𝐺 ∈ UHGraph → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ ( 𝒫 𝑉 ∖ { ∅ } ) ) |
| 4 |
3
|
adantr |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ ( 𝒫 𝑉 ∖ { ∅ } ) ) |
| 5 |
|
dmresss |
⊢ dom ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) ⊆ dom ( iEdg ‘ 𝐺 ) |
| 6 |
5
|
a1i |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → dom ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) ⊆ dom ( iEdg ‘ 𝐺 ) ) |
| 7 |
|
imadmres |
⊢ ( ( iEdg ‘ 𝐺 ) “ dom ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) ) = ( ( iEdg ‘ 𝐺 ) “ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) |
| 8 |
|
ffvelcdm |
⊢ ( ( ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ ( 𝒫 𝑉 ∖ { ∅ } ) ∧ 𝑦 ∈ dom ( iEdg ‘ 𝐺 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ) |
| 9 |
|
eldifsni |
⊢ ( ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ∈ ( 𝒫 𝑉 ∖ { ∅ } ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ≠ ∅ ) |
| 10 |
8 9
|
syl |
⊢ ( ( ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ ( 𝒫 𝑉 ∖ { ∅ } ) ∧ 𝑦 ∈ dom ( iEdg ‘ 𝐺 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ≠ ∅ ) |
| 11 |
10
|
ex |
⊢ ( ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ ( 𝒫 𝑉 ∖ { ∅ } ) → ( 𝑦 ∈ dom ( iEdg ‘ 𝐺 ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ≠ ∅ ) ) |
| 12 |
3 11
|
syl |
⊢ ( 𝐺 ∈ UHGraph → ( 𝑦 ∈ dom ( iEdg ‘ 𝐺 ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ≠ ∅ ) ) |
| 13 |
12
|
adantr |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( 𝑦 ∈ dom ( iEdg ‘ 𝐺 ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ≠ ∅ ) ) |
| 14 |
13
|
imp |
⊢ ( ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝑦 ∈ dom ( iEdg ‘ 𝐺 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ≠ ∅ ) |
| 15 |
|
fvexd |
⊢ ( ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ⊆ 𝑆 → ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ∈ V ) |
| 16 |
|
id |
⊢ ( ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ⊆ 𝑆 → ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ⊆ 𝑆 ) |
| 17 |
15 16
|
elpwd |
⊢ ( ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ⊆ 𝑆 → ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ∈ 𝒫 𝑆 ) |
| 18 |
14 17
|
anim12ci |
⊢ ( ( ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝑦 ∈ dom ( iEdg ‘ 𝐺 ) ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ⊆ 𝑆 ) → ( ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ∈ 𝒫 𝑆 ∧ ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ≠ ∅ ) ) |
| 19 |
|
eldifsn |
⊢ ( ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ∈ ( 𝒫 𝑆 ∖ { ∅ } ) ↔ ( ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ∈ 𝒫 𝑆 ∧ ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ≠ ∅ ) ) |
| 20 |
18 19
|
sylibr |
⊢ ( ( ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝑦 ∈ dom ( iEdg ‘ 𝐺 ) ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ⊆ 𝑆 ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ∈ ( 𝒫 𝑆 ∖ { ∅ } ) ) |
| 21 |
20
|
ex |
⊢ ( ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) ∧ 𝑦 ∈ dom ( iEdg ‘ 𝐺 ) ) → ( ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ⊆ 𝑆 → ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ∈ ( 𝒫 𝑆 ∖ { ∅ } ) ) ) |
| 22 |
21
|
ralrimiva |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ∀ 𝑦 ∈ dom ( iEdg ‘ 𝐺 ) ( ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ⊆ 𝑆 → ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ∈ ( 𝒫 𝑆 ∖ { ∅ } ) ) ) |
| 23 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) = ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ) |
| 24 |
23
|
sseq1d |
⊢ ( 𝑥 = 𝑦 → ( ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 ↔ ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ⊆ 𝑆 ) ) |
| 25 |
24
|
ralrab |
⊢ ( ∀ 𝑦 ∈ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ∈ ( 𝒫 𝑆 ∖ { ∅ } ) ↔ ∀ 𝑦 ∈ dom ( iEdg ‘ 𝐺 ) ( ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ⊆ 𝑆 → ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ∈ ( 𝒫 𝑆 ∖ { ∅ } ) ) ) |
| 26 |
22 25
|
sylibr |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ∀ 𝑦 ∈ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ∈ ( 𝒫 𝑆 ∖ { ∅ } ) ) |
| 27 |
|
ffun |
⊢ ( ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ ( 𝒫 𝑉 ∖ { ∅ } ) → Fun ( iEdg ‘ 𝐺 ) ) |
| 28 |
|
ssrab2 |
⊢ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ⊆ dom ( iEdg ‘ 𝐺 ) |
| 29 |
27 28
|
jctir |
⊢ ( ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ ( 𝒫 𝑉 ∖ { ∅ } ) → ( Fun ( iEdg ‘ 𝐺 ) ∧ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ⊆ dom ( iEdg ‘ 𝐺 ) ) ) |
| 30 |
3 29
|
syl |
⊢ ( 𝐺 ∈ UHGraph → ( Fun ( iEdg ‘ 𝐺 ) ∧ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ⊆ dom ( iEdg ‘ 𝐺 ) ) ) |
| 31 |
30
|
adantr |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( Fun ( iEdg ‘ 𝐺 ) ∧ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ⊆ dom ( iEdg ‘ 𝐺 ) ) ) |
| 32 |
|
funimass4 |
⊢ ( ( Fun ( iEdg ‘ 𝐺 ) ∧ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ⊆ dom ( iEdg ‘ 𝐺 ) ) → ( ( ( iEdg ‘ 𝐺 ) “ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) ⊆ ( 𝒫 𝑆 ∖ { ∅ } ) ↔ ∀ 𝑦 ∈ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ∈ ( 𝒫 𝑆 ∖ { ∅ } ) ) ) |
| 33 |
31 32
|
syl |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( ( ( iEdg ‘ 𝐺 ) “ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) ⊆ ( 𝒫 𝑆 ∖ { ∅ } ) ↔ ∀ 𝑦 ∈ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ∈ ( 𝒫 𝑆 ∖ { ∅ } ) ) ) |
| 34 |
26 33
|
mpbird |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( ( iEdg ‘ 𝐺 ) “ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) ⊆ ( 𝒫 𝑆 ∖ { ∅ } ) ) |
| 35 |
7 34
|
eqsstrid |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( ( iEdg ‘ 𝐺 ) “ dom ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) ) ⊆ ( 𝒫 𝑆 ∖ { ∅ } ) ) |
| 36 |
4 6 35
|
fssrescdmd |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( ( iEdg ‘ 𝐺 ) ↾ dom ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) ) : dom ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) ⟶ ( 𝒫 𝑆 ∖ { ∅ } ) ) |
| 37 |
|
resdmres |
⊢ ( ( iEdg ‘ 𝐺 ) ↾ dom ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) ) = ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) |
| 38 |
37
|
eqcomi |
⊢ ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) = ( ( iEdg ‘ 𝐺 ) ↾ dom ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) ) |
| 39 |
38
|
feq1i |
⊢ ( ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) : dom ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) ⟶ ( 𝒫 𝑆 ∖ { ∅ } ) ↔ ( ( iEdg ‘ 𝐺 ) ↾ dom ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) ) : dom ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) ⟶ ( 𝒫 𝑆 ∖ { ∅ } ) ) |
| 40 |
36 39
|
sylibr |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) : dom ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) ⟶ ( 𝒫 𝑆 ∖ { ∅ } ) ) |
| 41 |
1 2
|
isubgriedg |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( iEdg ‘ ( 𝐺 ISubGr 𝑆 ) ) = ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) ) |
| 42 |
41
|
dmeqd |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → dom ( iEdg ‘ ( 𝐺 ISubGr 𝑆 ) ) = dom ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) ) |
| 43 |
1
|
isubgrvtx |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( Vtx ‘ ( 𝐺 ISubGr 𝑆 ) ) = 𝑆 ) |
| 44 |
43
|
pweqd |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → 𝒫 ( Vtx ‘ ( 𝐺 ISubGr 𝑆 ) ) = 𝒫 𝑆 ) |
| 45 |
44
|
difeq1d |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( 𝒫 ( Vtx ‘ ( 𝐺 ISubGr 𝑆 ) ) ∖ { ∅ } ) = ( 𝒫 𝑆 ∖ { ∅ } ) ) |
| 46 |
41 42 45
|
feq123d |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( ( iEdg ‘ ( 𝐺 ISubGr 𝑆 ) ) : dom ( iEdg ‘ ( 𝐺 ISubGr 𝑆 ) ) ⟶ ( 𝒫 ( Vtx ‘ ( 𝐺 ISubGr 𝑆 ) ) ∖ { ∅ } ) ↔ ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) : dom ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) ⟶ ( 𝒫 𝑆 ∖ { ∅ } ) ) ) |
| 47 |
40 46
|
mpbird |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( iEdg ‘ ( 𝐺 ISubGr 𝑆 ) ) : dom ( iEdg ‘ ( 𝐺 ISubGr 𝑆 ) ) ⟶ ( 𝒫 ( Vtx ‘ ( 𝐺 ISubGr 𝑆 ) ) ∖ { ∅ } ) ) |
| 48 |
|
ovexd |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( 𝐺 ISubGr 𝑆 ) ∈ V ) |
| 49 |
|
eqid |
⊢ ( Vtx ‘ ( 𝐺 ISubGr 𝑆 ) ) = ( Vtx ‘ ( 𝐺 ISubGr 𝑆 ) ) |
| 50 |
|
eqid |
⊢ ( iEdg ‘ ( 𝐺 ISubGr 𝑆 ) ) = ( iEdg ‘ ( 𝐺 ISubGr 𝑆 ) ) |
| 51 |
49 50
|
isuhgr |
⊢ ( ( 𝐺 ISubGr 𝑆 ) ∈ V → ( ( 𝐺 ISubGr 𝑆 ) ∈ UHGraph ↔ ( iEdg ‘ ( 𝐺 ISubGr 𝑆 ) ) : dom ( iEdg ‘ ( 𝐺 ISubGr 𝑆 ) ) ⟶ ( 𝒫 ( Vtx ‘ ( 𝐺 ISubGr 𝑆 ) ) ∖ { ∅ } ) ) ) |
| 52 |
48 51
|
syl |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( ( 𝐺 ISubGr 𝑆 ) ∈ UHGraph ↔ ( iEdg ‘ ( 𝐺 ISubGr 𝑆 ) ) : dom ( iEdg ‘ ( 𝐺 ISubGr 𝑆 ) ) ⟶ ( 𝒫 ( Vtx ‘ ( 𝐺 ISubGr 𝑆 ) ) ∖ { ∅ } ) ) ) |
| 53 |
47 52
|
mpbird |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉 ) → ( 𝐺 ISubGr 𝑆 ) ∈ UHGraph ) |