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