Step |
Hyp |
Ref |
Expression |
1 |
|
isubgredg.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
isubgredg.e |
⊢ 𝐸 = ( Edg ‘ 𝐺 ) |
3 |
|
isubgredg.h |
⊢ 𝐻 = ( 𝐺 ISubGr 𝑆 ) |
4 |
|
isubgredg.i |
⊢ 𝐼 = ( Edg ‘ 𝐻 ) |
5 |
3
|
fveq2i |
⊢ ( iEdg ‘ 𝐻 ) = ( iEdg ‘ ( 𝐺 ISubGr 𝑆 ) ) |
6 |
|
eqid |
⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 ) |
7 |
1 6
|
isubgriedg |
⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ) → ( iEdg ‘ ( 𝐺 ISubGr 𝑆 ) ) = ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) ) |
8 |
5 7
|
eqtrid |
⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ) → ( iEdg ‘ 𝐻 ) = ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) ) |
9 |
8
|
rneqd |
⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ) → ran ( iEdg ‘ 𝐻 ) = ran ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) ) |
10 |
|
resss |
⊢ ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) ⊆ ( iEdg ‘ 𝐺 ) |
11 |
|
rnss |
⊢ ( ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) ⊆ ( iEdg ‘ 𝐺 ) → ran ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) ⊆ ran ( iEdg ‘ 𝐺 ) ) |
12 |
10 11
|
mp1i |
⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ) → ran ( ( iEdg ‘ 𝐺 ) ↾ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ⊆ 𝑆 } ) ⊆ ran ( iEdg ‘ 𝐺 ) ) |
13 |
9 12
|
eqsstrd |
⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ) → ran ( iEdg ‘ 𝐻 ) ⊆ ran ( iEdg ‘ 𝐺 ) ) |
14 |
|
edgval |
⊢ ( Edg ‘ 𝐻 ) = ran ( iEdg ‘ 𝐻 ) |
15 |
4 14
|
eqtri |
⊢ 𝐼 = ran ( iEdg ‘ 𝐻 ) |
16 |
|
edgval |
⊢ ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 ) |
17 |
2 16
|
eqtri |
⊢ 𝐸 = ran ( iEdg ‘ 𝐺 ) |
18 |
13 15 17
|
3sstr4g |
⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉 ) → 𝐼 ⊆ 𝐸 ) |