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 e. UHGraph /\ S C_ V ) -> ( iEdg ` ( G ISubGr S ) ) = ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) ) |
8 |
5 7
|
eqtrid |
|- ( ( G e. UHGraph /\ S C_ V ) -> ( iEdg ` H ) = ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) ) |
9 |
8
|
rneqd |
|- ( ( G e. UHGraph /\ S C_ V ) -> ran ( iEdg ` H ) = ran ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) ) |
10 |
9
|
eleq2d |
|- ( ( G e. UHGraph /\ S C_ V ) -> ( K e. ran ( iEdg ` H ) <-> K e. ran ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) ) ) |
11 |
1 6
|
uhgrf |
|- ( G e. UHGraph -> ( iEdg ` G ) : dom ( iEdg ` G ) --> ( ~P V \ { (/) } ) ) |
12 |
11
|
adantr |
|- ( ( G e. UHGraph /\ S C_ V ) -> ( iEdg ` G ) : dom ( iEdg ` G ) --> ( ~P V \ { (/) } ) ) |
13 |
12
|
ffnd |
|- ( ( G e. UHGraph /\ S C_ V ) -> ( iEdg ` G ) Fn dom ( iEdg ` G ) ) |
14 |
|
ssrab2 |
|- { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } C_ dom ( iEdg ` G ) |
15 |
14
|
a1i |
|- ( ( G e. UHGraph /\ S C_ V ) -> { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } C_ dom ( iEdg ` G ) ) |
16 |
13 15
|
fnssresd |
|- ( ( G e. UHGraph /\ S C_ V ) -> ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) Fn { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) |
17 |
|
fvelrnb |
|- ( ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) Fn { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } -> ( K e. ran ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) <-> E. x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ( ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) ` x ) = K ) ) |
18 |
16 17
|
syl |
|- ( ( G e. UHGraph /\ S C_ V ) -> ( K e. ran ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) <-> E. x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ( ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) ` x ) = K ) ) |
19 |
|
fvres |
|- ( x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } -> ( ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) ` x ) = ( ( iEdg ` G ) ` x ) ) |
20 |
19
|
adantl |
|- ( ( ( G e. UHGraph /\ S C_ V ) /\ x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) -> ( ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) ` x ) = ( ( iEdg ` G ) ` x ) ) |
21 |
20
|
eqeq1d |
|- ( ( ( G e. UHGraph /\ S C_ V ) /\ x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) -> ( ( ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) ` x ) = K <-> ( ( iEdg ` G ) ` x ) = K ) ) |
22 |
|
fveq2 |
|- ( i = x -> ( ( iEdg ` G ) ` i ) = ( ( iEdg ` G ) ` x ) ) |
23 |
22
|
sseq1d |
|- ( i = x -> ( ( ( iEdg ` G ) ` i ) C_ S <-> ( ( iEdg ` G ) ` x ) C_ S ) ) |
24 |
23
|
elrab |
|- ( x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } <-> ( x e. dom ( iEdg ` G ) /\ ( ( iEdg ` G ) ` x ) C_ S ) ) |
25 |
6
|
uhgrfun |
|- ( G e. UHGraph -> Fun ( iEdg ` G ) ) |
26 |
25
|
adantr |
|- ( ( G e. UHGraph /\ S C_ V ) -> Fun ( iEdg ` G ) ) |
27 |
|
simpl |
|- ( ( x e. dom ( iEdg ` G ) /\ ( ( iEdg ` G ) ` x ) C_ S ) -> x e. dom ( iEdg ` G ) ) |
28 |
|
fvelrn |
|- ( ( Fun ( iEdg ` G ) /\ x e. dom ( iEdg ` G ) ) -> ( ( iEdg ` G ) ` x ) e. ran ( iEdg ` G ) ) |
29 |
26 27 28
|
syl2anr |
|- ( ( ( x e. dom ( iEdg ` G ) /\ ( ( iEdg ` G ) ` x ) C_ S ) /\ ( G e. UHGraph /\ S C_ V ) ) -> ( ( iEdg ` G ) ` x ) e. ran ( iEdg ` G ) ) |
30 |
|
simpr |
|- ( ( x e. dom ( iEdg ` G ) /\ ( ( iEdg ` G ) ` x ) C_ S ) -> ( ( iEdg ` G ) ` x ) C_ S ) |
31 |
30
|
adantr |
|- ( ( ( x e. dom ( iEdg ` G ) /\ ( ( iEdg ` G ) ` x ) C_ S ) /\ ( G e. UHGraph /\ S C_ V ) ) -> ( ( iEdg ` G ) ` x ) C_ S ) |
32 |
29 31
|
jca |
|- ( ( ( x e. dom ( iEdg ` G ) /\ ( ( iEdg ` G ) ` x ) C_ S ) /\ ( G e. UHGraph /\ S C_ V ) ) -> ( ( ( iEdg ` G ) ` x ) e. ran ( iEdg ` G ) /\ ( ( iEdg ` G ) ` x ) C_ S ) ) |
33 |
32
|
ex |
|- ( ( x e. dom ( iEdg ` G ) /\ ( ( iEdg ` G ) ` x ) C_ S ) -> ( ( G e. UHGraph /\ S C_ V ) -> ( ( ( iEdg ` G ) ` x ) e. ran ( iEdg ` G ) /\ ( ( iEdg ` G ) ` x ) C_ S ) ) ) |
34 |
24 33
|
sylbi |
|- ( x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } -> ( ( G e. UHGraph /\ S C_ V ) -> ( ( ( iEdg ` G ) ` x ) e. ran ( iEdg ` G ) /\ ( ( iEdg ` G ) ` x ) C_ S ) ) ) |
35 |
34
|
impcom |
|- ( ( ( G e. UHGraph /\ S C_ V ) /\ x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) -> ( ( ( iEdg ` G ) ` x ) e. ran ( iEdg ` G ) /\ ( ( iEdg ` G ) ` x ) C_ S ) ) |
36 |
|
eleq1 |
|- ( ( ( iEdg ` G ) ` x ) = K -> ( ( ( iEdg ` G ) ` x ) e. ran ( iEdg ` G ) <-> K e. ran ( iEdg ` G ) ) ) |
37 |
|
sseq1 |
|- ( ( ( iEdg ` G ) ` x ) = K -> ( ( ( iEdg ` G ) ` x ) C_ S <-> K C_ S ) ) |
38 |
36 37
|
anbi12d |
|- ( ( ( iEdg ` G ) ` x ) = K -> ( ( ( ( iEdg ` G ) ` x ) e. ran ( iEdg ` G ) /\ ( ( iEdg ` G ) ` x ) C_ S ) <-> ( K e. ran ( iEdg ` G ) /\ K C_ S ) ) ) |
39 |
35 38
|
syl5ibcom |
|- ( ( ( G e. UHGraph /\ S C_ V ) /\ x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) -> ( ( ( iEdg ` G ) ` x ) = K -> ( K e. ran ( iEdg ` G ) /\ K C_ S ) ) ) |
40 |
21 39
|
sylbid |
|- ( ( ( G e. UHGraph /\ S C_ V ) /\ x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) -> ( ( ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) ` x ) = K -> ( K e. ran ( iEdg ` G ) /\ K C_ S ) ) ) |
41 |
40
|
rexlimdva |
|- ( ( G e. UHGraph /\ S C_ V ) -> ( E. x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ( ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) ` x ) = K -> ( K e. ran ( iEdg ` G ) /\ K C_ S ) ) ) |
42 |
|
edgval |
|- ( Edg ` G ) = ran ( iEdg ` G ) |
43 |
42
|
eqcomi |
|- ran ( iEdg ` G ) = ( Edg ` G ) |
44 |
43
|
eleq2i |
|- ( K e. ran ( iEdg ` G ) <-> K e. ( Edg ` G ) ) |
45 |
6
|
edgiedgb |
|- ( Fun ( iEdg ` G ) -> ( K e. ( Edg ` G ) <-> E. x e. dom ( iEdg ` G ) K = ( ( iEdg ` G ) ` x ) ) ) |
46 |
44 45
|
bitrid |
|- ( Fun ( iEdg ` G ) -> ( K e. ran ( iEdg ` G ) <-> E. x e. dom ( iEdg ` G ) K = ( ( iEdg ` G ) ` x ) ) ) |
47 |
25 46
|
syl |
|- ( G e. UHGraph -> ( K e. ran ( iEdg ` G ) <-> E. x e. dom ( iEdg ` G ) K = ( ( iEdg ` G ) ` x ) ) ) |
48 |
47
|
adantr |
|- ( ( G e. UHGraph /\ S C_ V ) -> ( K e. ran ( iEdg ` G ) <-> E. x e. dom ( iEdg ` G ) K = ( ( iEdg ` G ) ` x ) ) ) |
49 |
|
simprl |
|- ( ( ( ( G e. UHGraph /\ S C_ V ) /\ K C_ S ) /\ ( x e. dom ( iEdg ` G ) /\ K = ( ( iEdg ` G ) ` x ) ) ) -> x e. dom ( iEdg ` G ) ) |
50 |
|
simpr |
|- ( ( x e. dom ( iEdg ` G ) /\ K = ( ( iEdg ` G ) ` x ) ) -> K = ( ( iEdg ` G ) ` x ) ) |
51 |
50
|
sseq1d |
|- ( ( x e. dom ( iEdg ` G ) /\ K = ( ( iEdg ` G ) ` x ) ) -> ( K C_ S <-> ( ( iEdg ` G ) ` x ) C_ S ) ) |
52 |
51
|
biimpcd |
|- ( K C_ S -> ( ( x e. dom ( iEdg ` G ) /\ K = ( ( iEdg ` G ) ` x ) ) -> ( ( iEdg ` G ) ` x ) C_ S ) ) |
53 |
52
|
adantl |
|- ( ( ( G e. UHGraph /\ S C_ V ) /\ K C_ S ) -> ( ( x e. dom ( iEdg ` G ) /\ K = ( ( iEdg ` G ) ` x ) ) -> ( ( iEdg ` G ) ` x ) C_ S ) ) |
54 |
53
|
imp |
|- ( ( ( ( G e. UHGraph /\ S C_ V ) /\ K C_ S ) /\ ( x e. dom ( iEdg ` G ) /\ K = ( ( iEdg ` G ) ` x ) ) ) -> ( ( iEdg ` G ) ` x ) C_ S ) |
55 |
49 54 24
|
sylanbrc |
|- ( ( ( ( G e. UHGraph /\ S C_ V ) /\ K C_ S ) /\ ( x e. dom ( iEdg ` G ) /\ K = ( ( iEdg ` G ) ` x ) ) ) -> x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) |
56 |
|
simpr |
|- ( ( ( ( ( G e. UHGraph /\ S C_ V ) /\ K C_ S ) /\ ( x e. dom ( iEdg ` G ) /\ K = ( ( iEdg ` G ) ` x ) ) ) /\ x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) -> x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) |
57 |
50
|
eqcomd |
|- ( ( x e. dom ( iEdg ` G ) /\ K = ( ( iEdg ` G ) ` x ) ) -> ( ( iEdg ` G ) ` x ) = K ) |
58 |
57
|
adantl |
|- ( ( ( ( G e. UHGraph /\ S C_ V ) /\ K C_ S ) /\ ( x e. dom ( iEdg ` G ) /\ K = ( ( iEdg ` G ) ` x ) ) ) -> ( ( iEdg ` G ) ` x ) = K ) |
59 |
19 58
|
sylan9eqr |
|- ( ( ( ( ( G e. UHGraph /\ S C_ V ) /\ K C_ S ) /\ ( x e. dom ( iEdg ` G ) /\ K = ( ( iEdg ` G ) ` x ) ) ) /\ x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) -> ( ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) ` x ) = K ) |
60 |
56 59
|
jca |
|- ( ( ( ( ( G e. UHGraph /\ S C_ V ) /\ K C_ S ) /\ ( x e. dom ( iEdg ` G ) /\ K = ( ( iEdg ` G ) ` x ) ) ) /\ x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) -> ( x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } /\ ( ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) ` x ) = K ) ) |
61 |
55 60
|
mpdan |
|- ( ( ( ( G e. UHGraph /\ S C_ V ) /\ K C_ S ) /\ ( x e. dom ( iEdg ` G ) /\ K = ( ( iEdg ` G ) ` x ) ) ) -> ( x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } /\ ( ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) ` x ) = K ) ) |
62 |
61
|
ex |
|- ( ( ( G e. UHGraph /\ S C_ V ) /\ K C_ S ) -> ( ( x e. dom ( iEdg ` G ) /\ K = ( ( iEdg ` G ) ` x ) ) -> ( x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } /\ ( ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) ` x ) = K ) ) ) |
63 |
62
|
eximdv |
|- ( ( ( G e. UHGraph /\ S C_ V ) /\ K C_ S ) -> ( E. x ( x e. dom ( iEdg ` G ) /\ K = ( ( iEdg ` G ) ` x ) ) -> E. x ( x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } /\ ( ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) ` x ) = K ) ) ) |
64 |
|
df-rex |
|- ( E. x e. dom ( iEdg ` G ) K = ( ( iEdg ` G ) ` x ) <-> E. x ( x e. dom ( iEdg ` G ) /\ K = ( ( iEdg ` G ) ` x ) ) ) |
65 |
|
df-rex |
|- ( E. x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ( ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) ` x ) = K <-> E. x ( x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } /\ ( ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) ` x ) = K ) ) |
66 |
63 64 65
|
3imtr4g |
|- ( ( ( G e. UHGraph /\ S C_ V ) /\ K C_ S ) -> ( E. x e. dom ( iEdg ` G ) K = ( ( iEdg ` G ) ` x ) -> E. x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ( ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) ` x ) = K ) ) |
67 |
66
|
ex |
|- ( ( G e. UHGraph /\ S C_ V ) -> ( K C_ S -> ( E. x e. dom ( iEdg ` G ) K = ( ( iEdg ` G ) ` x ) -> E. x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ( ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) ` x ) = K ) ) ) |
68 |
67
|
com23 |
|- ( ( G e. UHGraph /\ S C_ V ) -> ( E. x e. dom ( iEdg ` G ) K = ( ( iEdg ` G ) ` x ) -> ( K C_ S -> E. x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ( ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) ` x ) = K ) ) ) |
69 |
48 68
|
sylbid |
|- ( ( G e. UHGraph /\ S C_ V ) -> ( K e. ran ( iEdg ` G ) -> ( K C_ S -> E. x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ( ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) ` x ) = K ) ) ) |
70 |
69
|
impd |
|- ( ( G e. UHGraph /\ S C_ V ) -> ( ( K e. ran ( iEdg ` G ) /\ K C_ S ) -> E. x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ( ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) ` x ) = K ) ) |
71 |
41 70
|
impbid |
|- ( ( G e. UHGraph /\ S C_ V ) -> ( E. x e. { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ( ( ( iEdg ` G ) |` { i e. dom ( iEdg ` G ) | ( ( iEdg ` G ) ` i ) C_ S } ) ` x ) = K <-> ( K e. ran ( iEdg ` G ) /\ K C_ S ) ) ) |
72 |
10 18 71
|
3bitrd |
|- ( ( G e. UHGraph /\ S C_ V ) -> ( K e. ran ( iEdg ` H ) <-> ( K e. ran ( iEdg ` G ) /\ K C_ S ) ) ) |
73 |
|
edgval |
|- ( Edg ` H ) = ran ( iEdg ` H ) |
74 |
4 73
|
eqtri |
|- I = ran ( iEdg ` H ) |
75 |
74
|
eleq2i |
|- ( K e. I <-> K e. ran ( iEdg ` H ) ) |
76 |
2 42
|
eqtri |
|- E = ran ( iEdg ` G ) |
77 |
76
|
eleq2i |
|- ( K e. E <-> K e. ran ( iEdg ` G ) ) |
78 |
77
|
anbi1i |
|- ( ( K e. E /\ K C_ S ) <-> ( K e. ran ( iEdg ` G ) /\ K C_ S ) ) |
79 |
72 75 78
|
3bitr4g |
|- ( ( G e. UHGraph /\ S C_ V ) -> ( K e. I <-> ( K e. E /\ K C_ S ) ) ) |