Step |
Hyp |
Ref |
Expression |
1 |
|
uhgrspan.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
uhgrspan.e |
⊢ 𝐸 = ( iEdg ‘ 𝐺 ) |
3 |
|
uhgrspan.s |
⊢ ( 𝜑 → 𝑆 ∈ 𝑊 ) |
4 |
|
uhgrspan.q |
⊢ ( 𝜑 → ( Vtx ‘ 𝑆 ) = 𝑉 ) |
5 |
|
uhgrspan.r |
⊢ ( 𝜑 → ( iEdg ‘ 𝑆 ) = ( 𝐸 ↾ 𝐴 ) ) |
6 |
|
uhgrspan.g |
⊢ ( 𝜑 → 𝐺 ∈ UHGraph ) |
7 |
|
edgval |
⊢ ( Edg ‘ 𝑆 ) = ran ( iEdg ‘ 𝑆 ) |
8 |
7
|
eleq2i |
⊢ ( 𝑒 ∈ ( Edg ‘ 𝑆 ) ↔ 𝑒 ∈ ran ( iEdg ‘ 𝑆 ) ) |
9 |
2
|
uhgrfun |
⊢ ( 𝐺 ∈ UHGraph → Fun 𝐸 ) |
10 |
|
funres |
⊢ ( Fun 𝐸 → Fun ( 𝐸 ↾ 𝐴 ) ) |
11 |
6 9 10
|
3syl |
⊢ ( 𝜑 → Fun ( 𝐸 ↾ 𝐴 ) ) |
12 |
5
|
funeqd |
⊢ ( 𝜑 → ( Fun ( iEdg ‘ 𝑆 ) ↔ Fun ( 𝐸 ↾ 𝐴 ) ) ) |
13 |
11 12
|
mpbird |
⊢ ( 𝜑 → Fun ( iEdg ‘ 𝑆 ) ) |
14 |
|
elrnrexdmb |
⊢ ( Fun ( iEdg ‘ 𝑆 ) → ( 𝑒 ∈ ran ( iEdg ‘ 𝑆 ) ↔ ∃ 𝑖 ∈ dom ( iEdg ‘ 𝑆 ) 𝑒 = ( ( iEdg ‘ 𝑆 ) ‘ 𝑖 ) ) ) |
15 |
13 14
|
syl |
⊢ ( 𝜑 → ( 𝑒 ∈ ran ( iEdg ‘ 𝑆 ) ↔ ∃ 𝑖 ∈ dom ( iEdg ‘ 𝑆 ) 𝑒 = ( ( iEdg ‘ 𝑆 ) ‘ 𝑖 ) ) ) |
16 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ dom ( iEdg ‘ 𝑆 ) ) → ( iEdg ‘ 𝑆 ) = ( 𝐸 ↾ 𝐴 ) ) |
17 |
16
|
fveq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ dom ( iEdg ‘ 𝑆 ) ) → ( ( iEdg ‘ 𝑆 ) ‘ 𝑖 ) = ( ( 𝐸 ↾ 𝐴 ) ‘ 𝑖 ) ) |
18 |
5
|
dmeqd |
⊢ ( 𝜑 → dom ( iEdg ‘ 𝑆 ) = dom ( 𝐸 ↾ 𝐴 ) ) |
19 |
|
dmres |
⊢ dom ( 𝐸 ↾ 𝐴 ) = ( 𝐴 ∩ dom 𝐸 ) |
20 |
18 19
|
eqtrdi |
⊢ ( 𝜑 → dom ( iEdg ‘ 𝑆 ) = ( 𝐴 ∩ dom 𝐸 ) ) |
21 |
20
|
eleq2d |
⊢ ( 𝜑 → ( 𝑖 ∈ dom ( iEdg ‘ 𝑆 ) ↔ 𝑖 ∈ ( 𝐴 ∩ dom 𝐸 ) ) ) |
22 |
|
elinel1 |
⊢ ( 𝑖 ∈ ( 𝐴 ∩ dom 𝐸 ) → 𝑖 ∈ 𝐴 ) |
23 |
21 22
|
syl6bi |
⊢ ( 𝜑 → ( 𝑖 ∈ dom ( iEdg ‘ 𝑆 ) → 𝑖 ∈ 𝐴 ) ) |
24 |
23
|
imp |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ dom ( iEdg ‘ 𝑆 ) ) → 𝑖 ∈ 𝐴 ) |
25 |
24
|
fvresd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ dom ( iEdg ‘ 𝑆 ) ) → ( ( 𝐸 ↾ 𝐴 ) ‘ 𝑖 ) = ( 𝐸 ‘ 𝑖 ) ) |
26 |
17 25
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ dom ( iEdg ‘ 𝑆 ) ) → ( ( iEdg ‘ 𝑆 ) ‘ 𝑖 ) = ( 𝐸 ‘ 𝑖 ) ) |
27 |
|
elinel2 |
⊢ ( 𝑖 ∈ ( 𝐴 ∩ dom 𝐸 ) → 𝑖 ∈ dom 𝐸 ) |
28 |
21 27
|
syl6bi |
⊢ ( 𝜑 → ( 𝑖 ∈ dom ( iEdg ‘ 𝑆 ) → 𝑖 ∈ dom 𝐸 ) ) |
29 |
28
|
imp |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ dom ( iEdg ‘ 𝑆 ) ) → 𝑖 ∈ dom 𝐸 ) |
30 |
1 2
|
uhgrss |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑖 ∈ dom 𝐸 ) → ( 𝐸 ‘ 𝑖 ) ⊆ 𝑉 ) |
31 |
6 29 30
|
syl2an2r |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ dom ( iEdg ‘ 𝑆 ) ) → ( 𝐸 ‘ 𝑖 ) ⊆ 𝑉 ) |
32 |
4
|
pweqd |
⊢ ( 𝜑 → 𝒫 ( Vtx ‘ 𝑆 ) = 𝒫 𝑉 ) |
33 |
32
|
eleq2d |
⊢ ( 𝜑 → ( ( 𝐸 ‘ 𝑖 ) ∈ 𝒫 ( Vtx ‘ 𝑆 ) ↔ ( 𝐸 ‘ 𝑖 ) ∈ 𝒫 𝑉 ) ) |
34 |
33
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ dom ( iEdg ‘ 𝑆 ) ) → ( ( 𝐸 ‘ 𝑖 ) ∈ 𝒫 ( Vtx ‘ 𝑆 ) ↔ ( 𝐸 ‘ 𝑖 ) ∈ 𝒫 𝑉 ) ) |
35 |
|
fvex |
⊢ ( 𝐸 ‘ 𝑖 ) ∈ V |
36 |
35
|
elpw |
⊢ ( ( 𝐸 ‘ 𝑖 ) ∈ 𝒫 𝑉 ↔ ( 𝐸 ‘ 𝑖 ) ⊆ 𝑉 ) |
37 |
34 36
|
bitrdi |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ dom ( iEdg ‘ 𝑆 ) ) → ( ( 𝐸 ‘ 𝑖 ) ∈ 𝒫 ( Vtx ‘ 𝑆 ) ↔ ( 𝐸 ‘ 𝑖 ) ⊆ 𝑉 ) ) |
38 |
31 37
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ dom ( iEdg ‘ 𝑆 ) ) → ( 𝐸 ‘ 𝑖 ) ∈ 𝒫 ( Vtx ‘ 𝑆 ) ) |
39 |
26 38
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ dom ( iEdg ‘ 𝑆 ) ) → ( ( iEdg ‘ 𝑆 ) ‘ 𝑖 ) ∈ 𝒫 ( Vtx ‘ 𝑆 ) ) |
40 |
|
eleq1 |
⊢ ( 𝑒 = ( ( iEdg ‘ 𝑆 ) ‘ 𝑖 ) → ( 𝑒 ∈ 𝒫 ( Vtx ‘ 𝑆 ) ↔ ( ( iEdg ‘ 𝑆 ) ‘ 𝑖 ) ∈ 𝒫 ( Vtx ‘ 𝑆 ) ) ) |
41 |
39 40
|
syl5ibrcom |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ dom ( iEdg ‘ 𝑆 ) ) → ( 𝑒 = ( ( iEdg ‘ 𝑆 ) ‘ 𝑖 ) → 𝑒 ∈ 𝒫 ( Vtx ‘ 𝑆 ) ) ) |
42 |
41
|
rexlimdva |
⊢ ( 𝜑 → ( ∃ 𝑖 ∈ dom ( iEdg ‘ 𝑆 ) 𝑒 = ( ( iEdg ‘ 𝑆 ) ‘ 𝑖 ) → 𝑒 ∈ 𝒫 ( Vtx ‘ 𝑆 ) ) ) |
43 |
15 42
|
sylbid |
⊢ ( 𝜑 → ( 𝑒 ∈ ran ( iEdg ‘ 𝑆 ) → 𝑒 ∈ 𝒫 ( Vtx ‘ 𝑆 ) ) ) |
44 |
8 43
|
syl5bi |
⊢ ( 𝜑 → ( 𝑒 ∈ ( Edg ‘ 𝑆 ) → 𝑒 ∈ 𝒫 ( Vtx ‘ 𝑆 ) ) ) |
45 |
44
|
ssrdv |
⊢ ( 𝜑 → ( Edg ‘ 𝑆 ) ⊆ 𝒫 ( Vtx ‘ 𝑆 ) ) |