Step |
Hyp |
Ref |
Expression |
1 |
|
subgruhgredgd.v |
⊢ 𝑉 = ( Vtx ‘ 𝑆 ) |
2 |
|
subgruhgredgd.i |
⊢ 𝐼 = ( iEdg ‘ 𝑆 ) |
3 |
|
subgruhgredgd.g |
⊢ ( 𝜑 → 𝐺 ∈ UHGraph ) |
4 |
|
subgruhgredgd.s |
⊢ ( 𝜑 → 𝑆 SubGraph 𝐺 ) |
5 |
|
subgruhgredgd.x |
⊢ ( 𝜑 → 𝑋 ∈ dom 𝐼 ) |
6 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
7 |
|
eqid |
⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 ) |
8 |
|
eqid |
⊢ ( Edg ‘ 𝑆 ) = ( Edg ‘ 𝑆 ) |
9 |
1 6 2 7 8
|
subgrprop2 |
⊢ ( 𝑆 SubGraph 𝐺 → ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) |
10 |
4 9
|
syl |
⊢ ( 𝜑 → ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) |
11 |
|
simpr3 |
⊢ ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) |
12 |
|
subgruhgrfun |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺 ) → Fun ( iEdg ‘ 𝑆 ) ) |
13 |
3 4 12
|
syl2anc |
⊢ ( 𝜑 → Fun ( iEdg ‘ 𝑆 ) ) |
14 |
2
|
dmeqi |
⊢ dom 𝐼 = dom ( iEdg ‘ 𝑆 ) |
15 |
5 14
|
eleqtrdi |
⊢ ( 𝜑 → 𝑋 ∈ dom ( iEdg ‘ 𝑆 ) ) |
16 |
13 15
|
jca |
⊢ ( 𝜑 → ( Fun ( iEdg ‘ 𝑆 ) ∧ 𝑋 ∈ dom ( iEdg ‘ 𝑆 ) ) ) |
17 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → ( Fun ( iEdg ‘ 𝑆 ) ∧ 𝑋 ∈ dom ( iEdg ‘ 𝑆 ) ) ) |
18 |
2
|
fveq1i |
⊢ ( 𝐼 ‘ 𝑋 ) = ( ( iEdg ‘ 𝑆 ) ‘ 𝑋 ) |
19 |
|
fvelrn |
⊢ ( ( Fun ( iEdg ‘ 𝑆 ) ∧ 𝑋 ∈ dom ( iEdg ‘ 𝑆 ) ) → ( ( iEdg ‘ 𝑆 ) ‘ 𝑋 ) ∈ ran ( iEdg ‘ 𝑆 ) ) |
20 |
18 19
|
eqeltrid |
⊢ ( ( Fun ( iEdg ‘ 𝑆 ) ∧ 𝑋 ∈ dom ( iEdg ‘ 𝑆 ) ) → ( 𝐼 ‘ 𝑋 ) ∈ ran ( iEdg ‘ 𝑆 ) ) |
21 |
17 20
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → ( 𝐼 ‘ 𝑋 ) ∈ ran ( iEdg ‘ 𝑆 ) ) |
22 |
|
edgval |
⊢ ( Edg ‘ 𝑆 ) = ran ( iEdg ‘ 𝑆 ) |
23 |
21 22
|
eleqtrrdi |
⊢ ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → ( 𝐼 ‘ 𝑋 ) ∈ ( Edg ‘ 𝑆 ) ) |
24 |
11 23
|
sseldd |
⊢ ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → ( 𝐼 ‘ 𝑋 ) ∈ 𝒫 𝑉 ) |
25 |
7
|
uhgrfun |
⊢ ( 𝐺 ∈ UHGraph → Fun ( iEdg ‘ 𝐺 ) ) |
26 |
3 25
|
syl |
⊢ ( 𝜑 → Fun ( iEdg ‘ 𝐺 ) ) |
27 |
26
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → Fun ( iEdg ‘ 𝐺 ) ) |
28 |
|
simpr2 |
⊢ ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ) |
29 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → 𝑋 ∈ dom 𝐼 ) |
30 |
|
funssfv |
⊢ ( ( Fun ( iEdg ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑋 ) = ( 𝐼 ‘ 𝑋 ) ) |
31 |
30
|
eqcomd |
⊢ ( ( Fun ( iEdg ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ 𝑋 ∈ dom 𝐼 ) → ( 𝐼 ‘ 𝑋 ) = ( ( iEdg ‘ 𝐺 ) ‘ 𝑋 ) ) |
32 |
27 28 29 31
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → ( 𝐼 ‘ 𝑋 ) = ( ( iEdg ‘ 𝐺 ) ‘ 𝑋 ) ) |
33 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → 𝐺 ∈ UHGraph ) |
34 |
26
|
funfnd |
⊢ ( 𝜑 → ( iEdg ‘ 𝐺 ) Fn dom ( iEdg ‘ 𝐺 ) ) |
35 |
34
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → ( iEdg ‘ 𝐺 ) Fn dom ( iEdg ‘ 𝐺 ) ) |
36 |
|
subgreldmiedg |
⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝑋 ∈ dom ( iEdg ‘ 𝑆 ) ) → 𝑋 ∈ dom ( iEdg ‘ 𝐺 ) ) |
37 |
4 15 36
|
syl2anc |
⊢ ( 𝜑 → 𝑋 ∈ dom ( iEdg ‘ 𝐺 ) ) |
38 |
37
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → 𝑋 ∈ dom ( iEdg ‘ 𝐺 ) ) |
39 |
7
|
uhgrn0 |
⊢ ( ( 𝐺 ∈ UHGraph ∧ ( iEdg ‘ 𝐺 ) Fn dom ( iEdg ‘ 𝐺 ) ∧ 𝑋 ∈ dom ( iEdg ‘ 𝐺 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑋 ) ≠ ∅ ) |
40 |
33 35 38 39
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ 𝑋 ) ≠ ∅ ) |
41 |
32 40
|
eqnetrd |
⊢ ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → ( 𝐼 ‘ 𝑋 ) ≠ ∅ ) |
42 |
|
eldifsn |
⊢ ( ( 𝐼 ‘ 𝑋 ) ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ↔ ( ( 𝐼 ‘ 𝑋 ) ∈ 𝒫 𝑉 ∧ ( 𝐼 ‘ 𝑋 ) ≠ ∅ ) ) |
43 |
24 41 42
|
sylanbrc |
⊢ ( ( 𝜑 ∧ ( 𝑉 ⊆ ( Vtx ‘ 𝐺 ) ∧ 𝐼 ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 𝑉 ) ) → ( 𝐼 ‘ 𝑋 ) ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ) |
44 |
10 43
|
mpdan |
⊢ ( 𝜑 → ( 𝐼 ‘ 𝑋 ) ∈ ( 𝒫 𝑉 ∖ { ∅ } ) ) |