Step |
Hyp |
Ref |
Expression |
1 |
|
ushgredgedgloop.e |
|- E = ( Edg ` G ) |
2 |
|
ushgredgedgloop.i |
|- I = ( iEdg ` G ) |
3 |
|
ushgredgedgloop.a |
|- A = { i e. dom I | ( I ` i ) = { N } } |
4 |
|
ushgredgedgloop.b |
|- B = { e e. E | e = { N } } |
5 |
|
ushgredgedgloop.f |
|- F = ( x e. A |-> ( I ` x ) ) |
6 |
|
eqid |
|- ( Vtx ` G ) = ( Vtx ` G ) |
7 |
6 2
|
ushgrf |
|- ( G e. USHGraph -> I : dom I -1-1-> ( ~P ( Vtx ` G ) \ { (/) } ) ) |
8 |
7
|
adantr |
|- ( ( G e. USHGraph /\ N e. V ) -> I : dom I -1-1-> ( ~P ( Vtx ` G ) \ { (/) } ) ) |
9 |
|
ssrab2 |
|- { i e. dom I | ( I ` i ) = { N } } C_ dom I |
10 |
|
f1ores |
|- ( ( I : dom I -1-1-> ( ~P ( Vtx ` G ) \ { (/) } ) /\ { i e. dom I | ( I ` i ) = { N } } C_ dom I ) -> ( I |` { i e. dom I | ( I ` i ) = { N } } ) : { i e. dom I | ( I ` i ) = { N } } -1-1-onto-> ( I " { i e. dom I | ( I ` i ) = { N } } ) ) |
11 |
8 9 10
|
sylancl |
|- ( ( G e. USHGraph /\ N e. V ) -> ( I |` { i e. dom I | ( I ` i ) = { N } } ) : { i e. dom I | ( I ` i ) = { N } } -1-1-onto-> ( I " { i e. dom I | ( I ` i ) = { N } } ) ) |
12 |
3
|
a1i |
|- ( ( G e. USHGraph /\ N e. V ) -> A = { i e. dom I | ( I ` i ) = { N } } ) |
13 |
|
eqidd |
|- ( ( ( G e. USHGraph /\ N e. V ) /\ x e. A ) -> ( I ` x ) = ( I ` x ) ) |
14 |
12 13
|
mpteq12dva |
|- ( ( G e. USHGraph /\ N e. V ) -> ( x e. A |-> ( I ` x ) ) = ( x e. { i e. dom I | ( I ` i ) = { N } } |-> ( I ` x ) ) ) |
15 |
5 14
|
syl5eq |
|- ( ( G e. USHGraph /\ N e. V ) -> F = ( x e. { i e. dom I | ( I ` i ) = { N } } |-> ( I ` x ) ) ) |
16 |
|
f1f |
|- ( I : dom I -1-1-> ( ~P ( Vtx ` G ) \ { (/) } ) -> I : dom I --> ( ~P ( Vtx ` G ) \ { (/) } ) ) |
17 |
7 16
|
syl |
|- ( G e. USHGraph -> I : dom I --> ( ~P ( Vtx ` G ) \ { (/) } ) ) |
18 |
9
|
a1i |
|- ( G e. USHGraph -> { i e. dom I | ( I ` i ) = { N } } C_ dom I ) |
19 |
17 18
|
feqresmpt |
|- ( G e. USHGraph -> ( I |` { i e. dom I | ( I ` i ) = { N } } ) = ( x e. { i e. dom I | ( I ` i ) = { N } } |-> ( I ` x ) ) ) |
20 |
19
|
adantr |
|- ( ( G e. USHGraph /\ N e. V ) -> ( I |` { i e. dom I | ( I ` i ) = { N } } ) = ( x e. { i e. dom I | ( I ` i ) = { N } } |-> ( I ` x ) ) ) |
21 |
15 20
|
eqtr4d |
|- ( ( G e. USHGraph /\ N e. V ) -> F = ( I |` { i e. dom I | ( I ` i ) = { N } } ) ) |
22 |
|
ushgruhgr |
|- ( G e. USHGraph -> G e. UHGraph ) |
23 |
|
eqid |
|- ( iEdg ` G ) = ( iEdg ` G ) |
24 |
23
|
uhgrfun |
|- ( G e. UHGraph -> Fun ( iEdg ` G ) ) |
25 |
22 24
|
syl |
|- ( G e. USHGraph -> Fun ( iEdg ` G ) ) |
26 |
2
|
funeqi |
|- ( Fun I <-> Fun ( iEdg ` G ) ) |
27 |
25 26
|
sylibr |
|- ( G e. USHGraph -> Fun I ) |
28 |
27
|
adantr |
|- ( ( G e. USHGraph /\ N e. V ) -> Fun I ) |
29 |
|
dfimafn |
|- ( ( Fun I /\ { i e. dom I | ( I ` i ) = { N } } C_ dom I ) -> ( I " { i e. dom I | ( I ` i ) = { N } } ) = { e | E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = e } ) |
30 |
28 9 29
|
sylancl |
|- ( ( G e. USHGraph /\ N e. V ) -> ( I " { i e. dom I | ( I ` i ) = { N } } ) = { e | E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = e } ) |
31 |
|
fveqeq2 |
|- ( i = j -> ( ( I ` i ) = { N } <-> ( I ` j ) = { N } ) ) |
32 |
31
|
elrab |
|- ( j e. { i e. dom I | ( I ` i ) = { N } } <-> ( j e. dom I /\ ( I ` j ) = { N } ) ) |
33 |
|
simpl |
|- ( ( j e. dom I /\ ( I ` j ) = { N } ) -> j e. dom I ) |
34 |
|
fvelrn |
|- ( ( Fun I /\ j e. dom I ) -> ( I ` j ) e. ran I ) |
35 |
2
|
eqcomi |
|- ( iEdg ` G ) = I |
36 |
35
|
rneqi |
|- ran ( iEdg ` G ) = ran I |
37 |
34 36
|
eleqtrrdi |
|- ( ( Fun I /\ j e. dom I ) -> ( I ` j ) e. ran ( iEdg ` G ) ) |
38 |
28 33 37
|
syl2an |
|- ( ( ( G e. USHGraph /\ N e. V ) /\ ( j e. dom I /\ ( I ` j ) = { N } ) ) -> ( I ` j ) e. ran ( iEdg ` G ) ) |
39 |
38
|
3adant3 |
|- ( ( ( G e. USHGraph /\ N e. V ) /\ ( j e. dom I /\ ( I ` j ) = { N } ) /\ ( I ` j ) = f ) -> ( I ` j ) e. ran ( iEdg ` G ) ) |
40 |
|
eleq1 |
|- ( f = ( I ` j ) -> ( f e. ran ( iEdg ` G ) <-> ( I ` j ) e. ran ( iEdg ` G ) ) ) |
41 |
40
|
eqcoms |
|- ( ( I ` j ) = f -> ( f e. ran ( iEdg ` G ) <-> ( I ` j ) e. ran ( iEdg ` G ) ) ) |
42 |
41
|
3ad2ant3 |
|- ( ( ( G e. USHGraph /\ N e. V ) /\ ( j e. dom I /\ ( I ` j ) = { N } ) /\ ( I ` j ) = f ) -> ( f e. ran ( iEdg ` G ) <-> ( I ` j ) e. ran ( iEdg ` G ) ) ) |
43 |
39 42
|
mpbird |
|- ( ( ( G e. USHGraph /\ N e. V ) /\ ( j e. dom I /\ ( I ` j ) = { N } ) /\ ( I ` j ) = f ) -> f e. ran ( iEdg ` G ) ) |
44 |
|
edgval |
|- ( Edg ` G ) = ran ( iEdg ` G ) |
45 |
44
|
a1i |
|- ( G e. USHGraph -> ( Edg ` G ) = ran ( iEdg ` G ) ) |
46 |
1 45
|
syl5eq |
|- ( G e. USHGraph -> E = ran ( iEdg ` G ) ) |
47 |
46
|
eleq2d |
|- ( G e. USHGraph -> ( f e. E <-> f e. ran ( iEdg ` G ) ) ) |
48 |
47
|
adantr |
|- ( ( G e. USHGraph /\ N e. V ) -> ( f e. E <-> f e. ran ( iEdg ` G ) ) ) |
49 |
48
|
3ad2ant1 |
|- ( ( ( G e. USHGraph /\ N e. V ) /\ ( j e. dom I /\ ( I ` j ) = { N } ) /\ ( I ` j ) = f ) -> ( f e. E <-> f e. ran ( iEdg ` G ) ) ) |
50 |
43 49
|
mpbird |
|- ( ( ( G e. USHGraph /\ N e. V ) /\ ( j e. dom I /\ ( I ` j ) = { N } ) /\ ( I ` j ) = f ) -> f e. E ) |
51 |
|
eqeq1 |
|- ( ( I ` j ) = f -> ( ( I ` j ) = { N } <-> f = { N } ) ) |
52 |
51
|
biimpcd |
|- ( ( I ` j ) = { N } -> ( ( I ` j ) = f -> f = { N } ) ) |
53 |
52
|
adantl |
|- ( ( j e. dom I /\ ( I ` j ) = { N } ) -> ( ( I ` j ) = f -> f = { N } ) ) |
54 |
53
|
a1i |
|- ( ( G e. USHGraph /\ N e. V ) -> ( ( j e. dom I /\ ( I ` j ) = { N } ) -> ( ( I ` j ) = f -> f = { N } ) ) ) |
55 |
54
|
3imp |
|- ( ( ( G e. USHGraph /\ N e. V ) /\ ( j e. dom I /\ ( I ` j ) = { N } ) /\ ( I ` j ) = f ) -> f = { N } ) |
56 |
50 55
|
jca |
|- ( ( ( G e. USHGraph /\ N e. V ) /\ ( j e. dom I /\ ( I ` j ) = { N } ) /\ ( I ` j ) = f ) -> ( f e. E /\ f = { N } ) ) |
57 |
56
|
3exp |
|- ( ( G e. USHGraph /\ N e. V ) -> ( ( j e. dom I /\ ( I ` j ) = { N } ) -> ( ( I ` j ) = f -> ( f e. E /\ f = { N } ) ) ) ) |
58 |
32 57
|
syl5bi |
|- ( ( G e. USHGraph /\ N e. V ) -> ( j e. { i e. dom I | ( I ` i ) = { N } } -> ( ( I ` j ) = f -> ( f e. E /\ f = { N } ) ) ) ) |
59 |
58
|
rexlimdv |
|- ( ( G e. USHGraph /\ N e. V ) -> ( E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = f -> ( f e. E /\ f = { N } ) ) ) |
60 |
25
|
funfnd |
|- ( G e. USHGraph -> ( iEdg ` G ) Fn dom ( iEdg ` G ) ) |
61 |
|
fvelrnb |
|- ( ( iEdg ` G ) Fn dom ( iEdg ` G ) -> ( f e. ran ( iEdg ` G ) <-> E. j e. dom ( iEdg ` G ) ( ( iEdg ` G ) ` j ) = f ) ) |
62 |
60 61
|
syl |
|- ( G e. USHGraph -> ( f e. ran ( iEdg ` G ) <-> E. j e. dom ( iEdg ` G ) ( ( iEdg ` G ) ` j ) = f ) ) |
63 |
35
|
dmeqi |
|- dom ( iEdg ` G ) = dom I |
64 |
63
|
eleq2i |
|- ( j e. dom ( iEdg ` G ) <-> j e. dom I ) |
65 |
64
|
biimpi |
|- ( j e. dom ( iEdg ` G ) -> j e. dom I ) |
66 |
65
|
adantr |
|- ( ( j e. dom ( iEdg ` G ) /\ ( ( iEdg ` G ) ` j ) = f ) -> j e. dom I ) |
67 |
66
|
adantl |
|- ( ( ( G e. USHGraph /\ f = { N } ) /\ ( j e. dom ( iEdg ` G ) /\ ( ( iEdg ` G ) ` j ) = f ) ) -> j e. dom I ) |
68 |
35
|
fveq1i |
|- ( ( iEdg ` G ) ` j ) = ( I ` j ) |
69 |
68
|
eqeq2i |
|- ( f = ( ( iEdg ` G ) ` j ) <-> f = ( I ` j ) ) |
70 |
69
|
biimpi |
|- ( f = ( ( iEdg ` G ) ` j ) -> f = ( I ` j ) ) |
71 |
70
|
eqcoms |
|- ( ( ( iEdg ` G ) ` j ) = f -> f = ( I ` j ) ) |
72 |
71
|
eqeq1d |
|- ( ( ( iEdg ` G ) ` j ) = f -> ( f = { N } <-> ( I ` j ) = { N } ) ) |
73 |
72
|
biimpcd |
|- ( f = { N } -> ( ( ( iEdg ` G ) ` j ) = f -> ( I ` j ) = { N } ) ) |
74 |
73
|
adantl |
|- ( ( G e. USHGraph /\ f = { N } ) -> ( ( ( iEdg ` G ) ` j ) = f -> ( I ` j ) = { N } ) ) |
75 |
74
|
adantld |
|- ( ( G e. USHGraph /\ f = { N } ) -> ( ( j e. dom ( iEdg ` G ) /\ ( ( iEdg ` G ) ` j ) = f ) -> ( I ` j ) = { N } ) ) |
76 |
75
|
imp |
|- ( ( ( G e. USHGraph /\ f = { N } ) /\ ( j e. dom ( iEdg ` G ) /\ ( ( iEdg ` G ) ` j ) = f ) ) -> ( I ` j ) = { N } ) |
77 |
67 76
|
jca |
|- ( ( ( G e. USHGraph /\ f = { N } ) /\ ( j e. dom ( iEdg ` G ) /\ ( ( iEdg ` G ) ` j ) = f ) ) -> ( j e. dom I /\ ( I ` j ) = { N } ) ) |
78 |
77 32
|
sylibr |
|- ( ( ( G e. USHGraph /\ f = { N } ) /\ ( j e. dom ( iEdg ` G ) /\ ( ( iEdg ` G ) ` j ) = f ) ) -> j e. { i e. dom I | ( I ` i ) = { N } } ) |
79 |
68
|
eqeq1i |
|- ( ( ( iEdg ` G ) ` j ) = f <-> ( I ` j ) = f ) |
80 |
79
|
biimpi |
|- ( ( ( iEdg ` G ) ` j ) = f -> ( I ` j ) = f ) |
81 |
80
|
adantl |
|- ( ( j e. dom ( iEdg ` G ) /\ ( ( iEdg ` G ) ` j ) = f ) -> ( I ` j ) = f ) |
82 |
81
|
adantl |
|- ( ( ( G e. USHGraph /\ f = { N } ) /\ ( j e. dom ( iEdg ` G ) /\ ( ( iEdg ` G ) ` j ) = f ) ) -> ( I ` j ) = f ) |
83 |
78 82
|
jca |
|- ( ( ( G e. USHGraph /\ f = { N } ) /\ ( j e. dom ( iEdg ` G ) /\ ( ( iEdg ` G ) ` j ) = f ) ) -> ( j e. { i e. dom I | ( I ` i ) = { N } } /\ ( I ` j ) = f ) ) |
84 |
83
|
ex |
|- ( ( G e. USHGraph /\ f = { N } ) -> ( ( j e. dom ( iEdg ` G ) /\ ( ( iEdg ` G ) ` j ) = f ) -> ( j e. { i e. dom I | ( I ` i ) = { N } } /\ ( I ` j ) = f ) ) ) |
85 |
84
|
reximdv2 |
|- ( ( G e. USHGraph /\ f = { N } ) -> ( E. j e. dom ( iEdg ` G ) ( ( iEdg ` G ) ` j ) = f -> E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = f ) ) |
86 |
85
|
ex |
|- ( G e. USHGraph -> ( f = { N } -> ( E. j e. dom ( iEdg ` G ) ( ( iEdg ` G ) ` j ) = f -> E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = f ) ) ) |
87 |
86
|
com23 |
|- ( G e. USHGraph -> ( E. j e. dom ( iEdg ` G ) ( ( iEdg ` G ) ` j ) = f -> ( f = { N } -> E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = f ) ) ) |
88 |
62 87
|
sylbid |
|- ( G e. USHGraph -> ( f e. ran ( iEdg ` G ) -> ( f = { N } -> E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = f ) ) ) |
89 |
47 88
|
sylbid |
|- ( G e. USHGraph -> ( f e. E -> ( f = { N } -> E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = f ) ) ) |
90 |
89
|
impd |
|- ( G e. USHGraph -> ( ( f e. E /\ f = { N } ) -> E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = f ) ) |
91 |
90
|
adantr |
|- ( ( G e. USHGraph /\ N e. V ) -> ( ( f e. E /\ f = { N } ) -> E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = f ) ) |
92 |
59 91
|
impbid |
|- ( ( G e. USHGraph /\ N e. V ) -> ( E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = f <-> ( f e. E /\ f = { N } ) ) ) |
93 |
|
vex |
|- f e. _V |
94 |
|
eqeq2 |
|- ( e = f -> ( ( I ` j ) = e <-> ( I ` j ) = f ) ) |
95 |
94
|
rexbidv |
|- ( e = f -> ( E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = e <-> E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = f ) ) |
96 |
93 95
|
elab |
|- ( f e. { e | E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = e } <-> E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = f ) |
97 |
|
eqeq1 |
|- ( e = f -> ( e = { N } <-> f = { N } ) ) |
98 |
97 4
|
elrab2 |
|- ( f e. B <-> ( f e. E /\ f = { N } ) ) |
99 |
92 96 98
|
3bitr4g |
|- ( ( G e. USHGraph /\ N e. V ) -> ( f e. { e | E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = e } <-> f e. B ) ) |
100 |
99
|
eqrdv |
|- ( ( G e. USHGraph /\ N e. V ) -> { e | E. j e. { i e. dom I | ( I ` i ) = { N } } ( I ` j ) = e } = B ) |
101 |
30 100
|
eqtr2d |
|- ( ( G e. USHGraph /\ N e. V ) -> B = ( I " { i e. dom I | ( I ` i ) = { N } } ) ) |
102 |
21 12 101
|
f1oeq123d |
|- ( ( G e. USHGraph /\ N e. V ) -> ( F : A -1-1-onto-> B <-> ( I |` { i e. dom I | ( I ` i ) = { N } } ) : { i e. dom I | ( I ` i ) = { N } } -1-1-onto-> ( I " { i e. dom I | ( I ` i ) = { N } } ) ) ) |
103 |
11 102
|
mpbird |
|- ( ( G e. USHGraph /\ N e. V ) -> F : A -1-1-onto-> B ) |