Step |
Hyp |
Ref |
Expression |
1 |
|
isomushgr.v |
|- V = ( Vtx ` A ) |
2 |
|
isomushgr.w |
|- W = ( Vtx ` B ) |
3 |
|
isomushgr.e |
|- E = ( Edg ` A ) |
4 |
|
isomushgr.k |
|- K = ( Edg ` B ) |
5 |
|
eqid |
|- ( iEdg ` A ) = ( iEdg ` A ) |
6 |
|
eqid |
|- ( iEdg ` B ) = ( iEdg ` B ) |
7 |
1 2 5 6
|
isomgr |
|- ( ( A e. USHGraph /\ B e. USHGraph ) -> ( A IsomGr B <-> E. f ( f : V -1-1-onto-> W /\ E. h ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) ) ) |
8 |
|
fvex |
|- ( iEdg ` B ) e. _V |
9 |
|
vex |
|- h e. _V |
10 |
|
fvex |
|- ( iEdg ` A ) e. _V |
11 |
10
|
cnvex |
|- `' ( iEdg ` A ) e. _V |
12 |
9 11
|
coex |
|- ( h o. `' ( iEdg ` A ) ) e. _V |
13 |
8 12
|
coex |
|- ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) e. _V |
14 |
13
|
a1i |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) -> ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) e. _V ) |
15 |
2 6
|
ushgrf |
|- ( B e. USHGraph -> ( iEdg ` B ) : dom ( iEdg ` B ) -1-1-> ( ~P W \ { (/) } ) ) |
16 |
|
f1f1orn |
|- ( ( iEdg ` B ) : dom ( iEdg ` B ) -1-1-> ( ~P W \ { (/) } ) -> ( iEdg ` B ) : dom ( iEdg ` B ) -1-1-onto-> ran ( iEdg ` B ) ) |
17 |
15 16
|
syl |
|- ( B e. USHGraph -> ( iEdg ` B ) : dom ( iEdg ` B ) -1-1-onto-> ran ( iEdg ` B ) ) |
18 |
|
edgval |
|- ( Edg ` B ) = ran ( iEdg ` B ) |
19 |
4 18
|
eqtri |
|- K = ran ( iEdg ` B ) |
20 |
|
f1oeq3 |
|- ( K = ran ( iEdg ` B ) -> ( ( iEdg ` B ) : dom ( iEdg ` B ) -1-1-onto-> K <-> ( iEdg ` B ) : dom ( iEdg ` B ) -1-1-onto-> ran ( iEdg ` B ) ) ) |
21 |
19 20
|
ax-mp |
|- ( ( iEdg ` B ) : dom ( iEdg ` B ) -1-1-onto-> K <-> ( iEdg ` B ) : dom ( iEdg ` B ) -1-1-onto-> ran ( iEdg ` B ) ) |
22 |
17 21
|
sylibr |
|- ( B e. USHGraph -> ( iEdg ` B ) : dom ( iEdg ` B ) -1-1-onto-> K ) |
23 |
22
|
ad3antlr |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) -> ( iEdg ` B ) : dom ( iEdg ` B ) -1-1-onto-> K ) |
24 |
|
simprl |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) -> h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) ) |
25 |
1 5
|
ushgrf |
|- ( A e. USHGraph -> ( iEdg ` A ) : dom ( iEdg ` A ) -1-1-> ( ~P V \ { (/) } ) ) |
26 |
|
f1f1orn |
|- ( ( iEdg ` A ) : dom ( iEdg ` A ) -1-1-> ( ~P V \ { (/) } ) -> ( iEdg ` A ) : dom ( iEdg ` A ) -1-1-onto-> ran ( iEdg ` A ) ) |
27 |
25 26
|
syl |
|- ( A e. USHGraph -> ( iEdg ` A ) : dom ( iEdg ` A ) -1-1-onto-> ran ( iEdg ` A ) ) |
28 |
|
edgval |
|- ( Edg ` A ) = ran ( iEdg ` A ) |
29 |
3 28
|
eqtri |
|- E = ran ( iEdg ` A ) |
30 |
|
f1oeq3 |
|- ( E = ran ( iEdg ` A ) -> ( ( iEdg ` A ) : dom ( iEdg ` A ) -1-1-onto-> E <-> ( iEdg ` A ) : dom ( iEdg ` A ) -1-1-onto-> ran ( iEdg ` A ) ) ) |
31 |
29 30
|
ax-mp |
|- ( ( iEdg ` A ) : dom ( iEdg ` A ) -1-1-onto-> E <-> ( iEdg ` A ) : dom ( iEdg ` A ) -1-1-onto-> ran ( iEdg ` A ) ) |
32 |
27 31
|
sylibr |
|- ( A e. USHGraph -> ( iEdg ` A ) : dom ( iEdg ` A ) -1-1-onto-> E ) |
33 |
|
f1ocnv |
|- ( ( iEdg ` A ) : dom ( iEdg ` A ) -1-1-onto-> E -> `' ( iEdg ` A ) : E -1-1-onto-> dom ( iEdg ` A ) ) |
34 |
32 33
|
syl |
|- ( A e. USHGraph -> `' ( iEdg ` A ) : E -1-1-onto-> dom ( iEdg ` A ) ) |
35 |
34
|
ad3antrrr |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) -> `' ( iEdg ` A ) : E -1-1-onto-> dom ( iEdg ` A ) ) |
36 |
|
f1oco |
|- ( ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ `' ( iEdg ` A ) : E -1-1-onto-> dom ( iEdg ` A ) ) -> ( h o. `' ( iEdg ` A ) ) : E -1-1-onto-> dom ( iEdg ` B ) ) |
37 |
24 35 36
|
syl2anc |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) -> ( h o. `' ( iEdg ` A ) ) : E -1-1-onto-> dom ( iEdg ` B ) ) |
38 |
|
f1oco |
|- ( ( ( iEdg ` B ) : dom ( iEdg ` B ) -1-1-onto-> K /\ ( h o. `' ( iEdg ` A ) ) : E -1-1-onto-> dom ( iEdg ` B ) ) -> ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) : E -1-1-onto-> K ) |
39 |
23 37 38
|
syl2anc |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) -> ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) : E -1-1-onto-> K ) |
40 |
29
|
eleq2i |
|- ( e e. E <-> e e. ran ( iEdg ` A ) ) |
41 |
|
f1fn |
|- ( ( iEdg ` A ) : dom ( iEdg ` A ) -1-1-> ( ~P V \ { (/) } ) -> ( iEdg ` A ) Fn dom ( iEdg ` A ) ) |
42 |
25 41
|
syl |
|- ( A e. USHGraph -> ( iEdg ` A ) Fn dom ( iEdg ` A ) ) |
43 |
|
fvelrnb |
|- ( ( iEdg ` A ) Fn dom ( iEdg ` A ) -> ( e e. ran ( iEdg ` A ) <-> E. j e. dom ( iEdg ` A ) ( ( iEdg ` A ) ` j ) = e ) ) |
44 |
42 43
|
syl |
|- ( A e. USHGraph -> ( e e. ran ( iEdg ` A ) <-> E. j e. dom ( iEdg ` A ) ( ( iEdg ` A ) ` j ) = e ) ) |
45 |
44
|
ad3antrrr |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) -> ( e e. ran ( iEdg ` A ) <-> E. j e. dom ( iEdg ` A ) ( ( iEdg ` A ) ` j ) = e ) ) |
46 |
|
fveq2 |
|- ( i = j -> ( ( iEdg ` A ) ` i ) = ( ( iEdg ` A ) ` j ) ) |
47 |
46
|
imaeq2d |
|- ( i = j -> ( f " ( ( iEdg ` A ) ` i ) ) = ( f " ( ( iEdg ` A ) ` j ) ) ) |
48 |
|
2fveq3 |
|- ( i = j -> ( ( iEdg ` B ) ` ( h ` i ) ) = ( ( iEdg ` B ) ` ( h ` j ) ) ) |
49 |
47 48
|
eqeq12d |
|- ( i = j -> ( ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) <-> ( f " ( ( iEdg ` A ) ` j ) ) = ( ( iEdg ` B ) ` ( h ` j ) ) ) ) |
50 |
49
|
rspccv |
|- ( A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) -> ( j e. dom ( iEdg ` A ) -> ( f " ( ( iEdg ` A ) ` j ) ) = ( ( iEdg ` B ) ` ( h ` j ) ) ) ) |
51 |
50
|
ad2antll |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) -> ( j e. dom ( iEdg ` A ) -> ( f " ( ( iEdg ` A ) ` j ) ) = ( ( iEdg ` B ) ` ( h ` j ) ) ) ) |
52 |
51
|
imp |
|- ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) /\ j e. dom ( iEdg ` A ) ) -> ( f " ( ( iEdg ` A ) ` j ) ) = ( ( iEdg ` B ) ` ( h ` j ) ) ) |
53 |
|
coass |
|- ( ( ( iEdg ` B ) o. h ) o. `' ( iEdg ` A ) ) = ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) |
54 |
53
|
eqcomi |
|- ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) = ( ( ( iEdg ` B ) o. h ) o. `' ( iEdg ` A ) ) |
55 |
54
|
fveq1i |
|- ( ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) ` ( ( iEdg ` A ) ` j ) ) = ( ( ( ( iEdg ` B ) o. h ) o. `' ( iEdg ` A ) ) ` ( ( iEdg ` A ) ` j ) ) |
56 |
|
dff1o4 |
|- ( ( iEdg ` A ) : dom ( iEdg ` A ) -1-1-onto-> ran ( iEdg ` A ) <-> ( ( iEdg ` A ) Fn dom ( iEdg ` A ) /\ `' ( iEdg ` A ) Fn ran ( iEdg ` A ) ) ) |
57 |
27 56
|
sylib |
|- ( A e. USHGraph -> ( ( iEdg ` A ) Fn dom ( iEdg ` A ) /\ `' ( iEdg ` A ) Fn ran ( iEdg ` A ) ) ) |
58 |
57
|
simprd |
|- ( A e. USHGraph -> `' ( iEdg ` A ) Fn ran ( iEdg ` A ) ) |
59 |
58
|
ad4antr |
|- ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) /\ j e. dom ( iEdg ` A ) ) -> `' ( iEdg ` A ) Fn ran ( iEdg ` A ) ) |
60 |
|
f1of |
|- ( ( iEdg ` A ) : dom ( iEdg ` A ) -1-1-onto-> ran ( iEdg ` A ) -> ( iEdg ` A ) : dom ( iEdg ` A ) --> ran ( iEdg ` A ) ) |
61 |
27 60
|
syl |
|- ( A e. USHGraph -> ( iEdg ` A ) : dom ( iEdg ` A ) --> ran ( iEdg ` A ) ) |
62 |
61
|
ad3antrrr |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) -> ( iEdg ` A ) : dom ( iEdg ` A ) --> ran ( iEdg ` A ) ) |
63 |
62
|
ffvelrnda |
|- ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) /\ j e. dom ( iEdg ` A ) ) -> ( ( iEdg ` A ) ` j ) e. ran ( iEdg ` A ) ) |
64 |
|
fvco2 |
|- ( ( `' ( iEdg ` A ) Fn ran ( iEdg ` A ) /\ ( ( iEdg ` A ) ` j ) e. ran ( iEdg ` A ) ) -> ( ( ( ( iEdg ` B ) o. h ) o. `' ( iEdg ` A ) ) ` ( ( iEdg ` A ) ` j ) ) = ( ( ( iEdg ` B ) o. h ) ` ( `' ( iEdg ` A ) ` ( ( iEdg ` A ) ` j ) ) ) ) |
65 |
59 63 64
|
syl2anc |
|- ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) /\ j e. dom ( iEdg ` A ) ) -> ( ( ( ( iEdg ` B ) o. h ) o. `' ( iEdg ` A ) ) ` ( ( iEdg ` A ) ` j ) ) = ( ( ( iEdg ` B ) o. h ) ` ( `' ( iEdg ` A ) ` ( ( iEdg ` A ) ` j ) ) ) ) |
66 |
32
|
ad3antrrr |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) -> ( iEdg ` A ) : dom ( iEdg ` A ) -1-1-onto-> E ) |
67 |
|
f1ocnvfv1 |
|- ( ( ( iEdg ` A ) : dom ( iEdg ` A ) -1-1-onto-> E /\ j e. dom ( iEdg ` A ) ) -> ( `' ( iEdg ` A ) ` ( ( iEdg ` A ) ` j ) ) = j ) |
68 |
66 67
|
sylan |
|- ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) /\ j e. dom ( iEdg ` A ) ) -> ( `' ( iEdg ` A ) ` ( ( iEdg ` A ) ` j ) ) = j ) |
69 |
68
|
fveq2d |
|- ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) /\ j e. dom ( iEdg ` A ) ) -> ( ( ( iEdg ` B ) o. h ) ` ( `' ( iEdg ` A ) ` ( ( iEdg ` A ) ` j ) ) ) = ( ( ( iEdg ` B ) o. h ) ` j ) ) |
70 |
|
f1ofn |
|- ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) -> h Fn dom ( iEdg ` A ) ) |
71 |
70
|
ad2antrl |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) -> h Fn dom ( iEdg ` A ) ) |
72 |
|
fvco2 |
|- ( ( h Fn dom ( iEdg ` A ) /\ j e. dom ( iEdg ` A ) ) -> ( ( ( iEdg ` B ) o. h ) ` j ) = ( ( iEdg ` B ) ` ( h ` j ) ) ) |
73 |
71 72
|
sylan |
|- ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) /\ j e. dom ( iEdg ` A ) ) -> ( ( ( iEdg ` B ) o. h ) ` j ) = ( ( iEdg ` B ) ` ( h ` j ) ) ) |
74 |
65 69 73
|
3eqtrd |
|- ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) /\ j e. dom ( iEdg ` A ) ) -> ( ( ( ( iEdg ` B ) o. h ) o. `' ( iEdg ` A ) ) ` ( ( iEdg ` A ) ` j ) ) = ( ( iEdg ` B ) ` ( h ` j ) ) ) |
75 |
55 74
|
eqtr2id |
|- ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) /\ j e. dom ( iEdg ` A ) ) -> ( ( iEdg ` B ) ` ( h ` j ) ) = ( ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) ` ( ( iEdg ` A ) ` j ) ) ) |
76 |
75
|
ad2antrr |
|- ( ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) /\ j e. dom ( iEdg ` A ) ) /\ ( f " ( ( iEdg ` A ) ` j ) ) = ( ( iEdg ` B ) ` ( h ` j ) ) ) /\ ( ( iEdg ` A ) ` j ) = e ) -> ( ( iEdg ` B ) ` ( h ` j ) ) = ( ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) ` ( ( iEdg ` A ) ` j ) ) ) |
77 |
|
imaeq2 |
|- ( e = ( ( iEdg ` A ) ` j ) -> ( f " e ) = ( f " ( ( iEdg ` A ) ` j ) ) ) |
78 |
77
|
eqcoms |
|- ( ( ( iEdg ` A ) ` j ) = e -> ( f " e ) = ( f " ( ( iEdg ` A ) ` j ) ) ) |
79 |
|
simpr |
|- ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) /\ j e. dom ( iEdg ` A ) ) /\ ( f " ( ( iEdg ` A ) ` j ) ) = ( ( iEdg ` B ) ` ( h ` j ) ) ) -> ( f " ( ( iEdg ` A ) ` j ) ) = ( ( iEdg ` B ) ` ( h ` j ) ) ) |
80 |
78 79
|
sylan9eqr |
|- ( ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) /\ j e. dom ( iEdg ` A ) ) /\ ( f " ( ( iEdg ` A ) ` j ) ) = ( ( iEdg ` B ) ` ( h ` j ) ) ) /\ ( ( iEdg ` A ) ` j ) = e ) -> ( f " e ) = ( ( iEdg ` B ) ` ( h ` j ) ) ) |
81 |
|
fveq2 |
|- ( e = ( ( iEdg ` A ) ` j ) -> ( ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) ` e ) = ( ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) ` ( ( iEdg ` A ) ` j ) ) ) |
82 |
81
|
eqcoms |
|- ( ( ( iEdg ` A ) ` j ) = e -> ( ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) ` e ) = ( ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) ` ( ( iEdg ` A ) ` j ) ) ) |
83 |
82
|
adantl |
|- ( ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) /\ j e. dom ( iEdg ` A ) ) /\ ( f " ( ( iEdg ` A ) ` j ) ) = ( ( iEdg ` B ) ` ( h ` j ) ) ) /\ ( ( iEdg ` A ) ` j ) = e ) -> ( ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) ` e ) = ( ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) ` ( ( iEdg ` A ) ` j ) ) ) |
84 |
76 80 83
|
3eqtr4d |
|- ( ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) /\ j e. dom ( iEdg ` A ) ) /\ ( f " ( ( iEdg ` A ) ` j ) ) = ( ( iEdg ` B ) ` ( h ` j ) ) ) /\ ( ( iEdg ` A ) ` j ) = e ) -> ( f " e ) = ( ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) ` e ) ) |
85 |
84
|
ex |
|- ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) /\ j e. dom ( iEdg ` A ) ) /\ ( f " ( ( iEdg ` A ) ` j ) ) = ( ( iEdg ` B ) ` ( h ` j ) ) ) -> ( ( ( iEdg ` A ) ` j ) = e -> ( f " e ) = ( ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) ` e ) ) ) |
86 |
52 85
|
mpdan |
|- ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) /\ j e. dom ( iEdg ` A ) ) -> ( ( ( iEdg ` A ) ` j ) = e -> ( f " e ) = ( ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) ` e ) ) ) |
87 |
86
|
rexlimdva |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) -> ( E. j e. dom ( iEdg ` A ) ( ( iEdg ` A ) ` j ) = e -> ( f " e ) = ( ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) ` e ) ) ) |
88 |
45 87
|
sylbid |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) -> ( e e. ran ( iEdg ` A ) -> ( f " e ) = ( ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) ` e ) ) ) |
89 |
40 88
|
syl5bi |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) -> ( e e. E -> ( f " e ) = ( ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) ` e ) ) ) |
90 |
89
|
ralrimiv |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) -> A. e e. E ( f " e ) = ( ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) ` e ) ) |
91 |
39 90
|
jca |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) -> ( ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) ` e ) ) ) |
92 |
|
f1oeq1 |
|- ( g = ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) -> ( g : E -1-1-onto-> K <-> ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) : E -1-1-onto-> K ) ) |
93 |
|
fveq1 |
|- ( g = ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) -> ( g ` e ) = ( ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) ` e ) ) |
94 |
93
|
eqeq2d |
|- ( g = ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) -> ( ( f " e ) = ( g ` e ) <-> ( f " e ) = ( ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) ` e ) ) ) |
95 |
94
|
ralbidv |
|- ( g = ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) -> ( A. e e. E ( f " e ) = ( g ` e ) <-> A. e e. E ( f " e ) = ( ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) ` e ) ) ) |
96 |
92 95
|
anbi12d |
|- ( g = ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) -> ( ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) <-> ( ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( ( ( iEdg ` B ) o. ( h o. `' ( iEdg ` A ) ) ) ` e ) ) ) ) |
97 |
14 91 96
|
spcedv |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) -> E. g ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) |
98 |
97
|
ex |
|- ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) -> ( ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) -> E. g ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) ) |
99 |
98
|
exlimdv |
|- ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) -> ( E. h ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) -> E. g ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) ) |
100 |
8
|
cnvex |
|- `' ( iEdg ` B ) e. _V |
101 |
|
vex |
|- g e. _V |
102 |
101 10
|
coex |
|- ( g o. ( iEdg ` A ) ) e. _V |
103 |
100 102
|
coex |
|- ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) e. _V |
104 |
103
|
a1i |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) -> ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) e. _V ) |
105 |
17
|
ad3antlr |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) -> ( iEdg ` B ) : dom ( iEdg ` B ) -1-1-onto-> ran ( iEdg ` B ) ) |
106 |
|
f1ocnv |
|- ( ( iEdg ` B ) : dom ( iEdg ` B ) -1-1-onto-> ran ( iEdg ` B ) -> `' ( iEdg ` B ) : ran ( iEdg ` B ) -1-1-onto-> dom ( iEdg ` B ) ) |
107 |
105 106
|
syl |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) -> `' ( iEdg ` B ) : ran ( iEdg ` B ) -1-1-onto-> dom ( iEdg ` B ) ) |
108 |
|
f1oeq23 |
|- ( ( E = ran ( iEdg ` A ) /\ K = ran ( iEdg ` B ) ) -> ( g : E -1-1-onto-> K <-> g : ran ( iEdg ` A ) -1-1-onto-> ran ( iEdg ` B ) ) ) |
109 |
29 19 108
|
mp2an |
|- ( g : E -1-1-onto-> K <-> g : ran ( iEdg ` A ) -1-1-onto-> ran ( iEdg ` B ) ) |
110 |
109
|
biimpi |
|- ( g : E -1-1-onto-> K -> g : ran ( iEdg ` A ) -1-1-onto-> ran ( iEdg ` B ) ) |
111 |
110
|
ad2antrl |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) -> g : ran ( iEdg ` A ) -1-1-onto-> ran ( iEdg ` B ) ) |
112 |
27
|
ad3antrrr |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) -> ( iEdg ` A ) : dom ( iEdg ` A ) -1-1-onto-> ran ( iEdg ` A ) ) |
113 |
|
f1oco |
|- ( ( g : ran ( iEdg ` A ) -1-1-onto-> ran ( iEdg ` B ) /\ ( iEdg ` A ) : dom ( iEdg ` A ) -1-1-onto-> ran ( iEdg ` A ) ) -> ( g o. ( iEdg ` A ) ) : dom ( iEdg ` A ) -1-1-onto-> ran ( iEdg ` B ) ) |
114 |
111 112 113
|
syl2anc |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) -> ( g o. ( iEdg ` A ) ) : dom ( iEdg ` A ) -1-1-onto-> ran ( iEdg ` B ) ) |
115 |
|
f1oco |
|- ( ( `' ( iEdg ` B ) : ran ( iEdg ` B ) -1-1-onto-> dom ( iEdg ` B ) /\ ( g o. ( iEdg ` A ) ) : dom ( iEdg ` A ) -1-1-onto-> ran ( iEdg ` B ) ) -> ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) ) |
116 |
107 114 115
|
syl2anc |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) -> ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) ) |
117 |
61
|
ad2antrr |
|- ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) -> ( iEdg ` A ) : dom ( iEdg ` A ) --> ran ( iEdg ` A ) ) |
118 |
117
|
ffund |
|- ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) -> Fun ( iEdg ` A ) ) |
119 |
118
|
adantr |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) -> Fun ( iEdg ` A ) ) |
120 |
|
fvelrn |
|- ( ( Fun ( iEdg ` A ) /\ i e. dom ( iEdg ` A ) ) -> ( ( iEdg ` A ) ` i ) e. ran ( iEdg ` A ) ) |
121 |
119 120
|
sylan |
|- ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) -> ( ( iEdg ` A ) ` i ) e. ran ( iEdg ` A ) ) |
122 |
29
|
raleqi |
|- ( A. e e. E ( f " e ) = ( g ` e ) <-> A. e e. ran ( iEdg ` A ) ( f " e ) = ( g ` e ) ) |
123 |
122
|
biimpi |
|- ( A. e e. E ( f " e ) = ( g ` e ) -> A. e e. ran ( iEdg ` A ) ( f " e ) = ( g ` e ) ) |
124 |
123
|
ad2antll |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) -> A. e e. ran ( iEdg ` A ) ( f " e ) = ( g ` e ) ) |
125 |
124
|
adantr |
|- ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) -> A. e e. ran ( iEdg ` A ) ( f " e ) = ( g ` e ) ) |
126 |
|
imaeq2 |
|- ( e = ( ( iEdg ` A ) ` i ) -> ( f " e ) = ( f " ( ( iEdg ` A ) ` i ) ) ) |
127 |
|
fveq2 |
|- ( e = ( ( iEdg ` A ) ` i ) -> ( g ` e ) = ( g ` ( ( iEdg ` A ) ` i ) ) ) |
128 |
126 127
|
eqeq12d |
|- ( e = ( ( iEdg ` A ) ` i ) -> ( ( f " e ) = ( g ` e ) <-> ( f " ( ( iEdg ` A ) ` i ) ) = ( g ` ( ( iEdg ` A ) ` i ) ) ) ) |
129 |
128
|
rspccva |
|- ( ( A. e e. ran ( iEdg ` A ) ( f " e ) = ( g ` e ) /\ ( ( iEdg ` A ) ` i ) e. ran ( iEdg ` A ) ) -> ( f " ( ( iEdg ` A ) ` i ) ) = ( g ` ( ( iEdg ` A ) ` i ) ) ) |
130 |
125 129
|
sylan |
|- ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) /\ ( ( iEdg ` A ) ` i ) e. ran ( iEdg ` A ) ) -> ( f " ( ( iEdg ` A ) ` i ) ) = ( g ` ( ( iEdg ` A ) ` i ) ) ) |
131 |
|
feq3 |
|- ( E = ran ( iEdg ` A ) -> ( ( iEdg ` A ) : dom ( iEdg ` A ) --> E <-> ( iEdg ` A ) : dom ( iEdg ` A ) --> ran ( iEdg ` A ) ) ) |
132 |
29 131
|
ax-mp |
|- ( ( iEdg ` A ) : dom ( iEdg ` A ) --> E <-> ( iEdg ` A ) : dom ( iEdg ` A ) --> ran ( iEdg ` A ) ) |
133 |
61 132
|
sylibr |
|- ( A e. USHGraph -> ( iEdg ` A ) : dom ( iEdg ` A ) --> E ) |
134 |
133
|
ad2antrr |
|- ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) -> ( iEdg ` A ) : dom ( iEdg ` A ) --> E ) |
135 |
|
f1ofn |
|- ( g : E -1-1-onto-> K -> g Fn E ) |
136 |
135
|
adantr |
|- ( ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) -> g Fn E ) |
137 |
134 136
|
anim12ci |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) -> ( g Fn E /\ ( iEdg ` A ) : dom ( iEdg ` A ) --> E ) ) |
138 |
137
|
ad2antrr |
|- ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) /\ ( ( iEdg ` A ) ` i ) e. ran ( iEdg ` A ) ) -> ( g Fn E /\ ( iEdg ` A ) : dom ( iEdg ` A ) --> E ) ) |
139 |
|
fnfco |
|- ( ( g Fn E /\ ( iEdg ` A ) : dom ( iEdg ` A ) --> E ) -> ( g o. ( iEdg ` A ) ) Fn dom ( iEdg ` A ) ) |
140 |
138 139
|
syl |
|- ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) /\ ( ( iEdg ` A ) ` i ) e. ran ( iEdg ` A ) ) -> ( g o. ( iEdg ` A ) ) Fn dom ( iEdg ` A ) ) |
141 |
|
simplr |
|- ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) /\ ( ( iEdg ` A ) ` i ) e. ran ( iEdg ` A ) ) -> i e. dom ( iEdg ` A ) ) |
142 |
|
fvco2 |
|- ( ( ( g o. ( iEdg ` A ) ) Fn dom ( iEdg ` A ) /\ i e. dom ( iEdg ` A ) ) -> ( ( ( _I |` ran ( iEdg ` B ) ) o. ( g o. ( iEdg ` A ) ) ) ` i ) = ( ( _I |` ran ( iEdg ` B ) ) ` ( ( g o. ( iEdg ` A ) ) ` i ) ) ) |
143 |
140 141 142
|
syl2anc |
|- ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) /\ ( ( iEdg ` A ) ` i ) e. ran ( iEdg ` A ) ) -> ( ( ( _I |` ran ( iEdg ` B ) ) o. ( g o. ( iEdg ` A ) ) ) ` i ) = ( ( _I |` ran ( iEdg ` B ) ) ` ( ( g o. ( iEdg ` A ) ) ` i ) ) ) |
144 |
|
f1of |
|- ( ( iEdg ` B ) : dom ( iEdg ` B ) -1-1-onto-> K -> ( iEdg ` B ) : dom ( iEdg ` B ) --> K ) |
145 |
22 144
|
syl |
|- ( B e. USHGraph -> ( iEdg ` B ) : dom ( iEdg ` B ) --> K ) |
146 |
145
|
ffund |
|- ( B e. USHGraph -> Fun ( iEdg ` B ) ) |
147 |
|
funcocnv2 |
|- ( Fun ( iEdg ` B ) -> ( ( iEdg ` B ) o. `' ( iEdg ` B ) ) = ( _I |` ran ( iEdg ` B ) ) ) |
148 |
146 147
|
syl |
|- ( B e. USHGraph -> ( ( iEdg ` B ) o. `' ( iEdg ` B ) ) = ( _I |` ran ( iEdg ` B ) ) ) |
149 |
148
|
eqcomd |
|- ( B e. USHGraph -> ( _I |` ran ( iEdg ` B ) ) = ( ( iEdg ` B ) o. `' ( iEdg ` B ) ) ) |
150 |
149
|
ad5antlr |
|- ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) /\ ( ( iEdg ` A ) ` i ) e. ran ( iEdg ` A ) ) -> ( _I |` ran ( iEdg ` B ) ) = ( ( iEdg ` B ) o. `' ( iEdg ` B ) ) ) |
151 |
150
|
coeq1d |
|- ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) /\ ( ( iEdg ` A ) ` i ) e. ran ( iEdg ` A ) ) -> ( ( _I |` ran ( iEdg ` B ) ) o. ( g o. ( iEdg ` A ) ) ) = ( ( ( iEdg ` B ) o. `' ( iEdg ` B ) ) o. ( g o. ( iEdg ` A ) ) ) ) |
152 |
151
|
fveq1d |
|- ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) /\ ( ( iEdg ` A ) ` i ) e. ran ( iEdg ` A ) ) -> ( ( ( _I |` ran ( iEdg ` B ) ) o. ( g o. ( iEdg ` A ) ) ) ` i ) = ( ( ( ( iEdg ` B ) o. `' ( iEdg ` B ) ) o. ( g o. ( iEdg ` A ) ) ) ` i ) ) |
153 |
|
coass |
|- ( ( ( iEdg ` B ) o. `' ( iEdg ` B ) ) o. ( g o. ( iEdg ` A ) ) ) = ( ( iEdg ` B ) o. ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) ) |
154 |
153
|
fveq1i |
|- ( ( ( ( iEdg ` B ) o. `' ( iEdg ` B ) ) o. ( g o. ( iEdg ` A ) ) ) ` i ) = ( ( ( iEdg ` B ) o. ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) ) ` i ) |
155 |
152 154
|
eqtrdi |
|- ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) /\ ( ( iEdg ` A ) ` i ) e. ran ( iEdg ` A ) ) -> ( ( ( _I |` ran ( iEdg ` B ) ) o. ( g o. ( iEdg ` A ) ) ) ` i ) = ( ( ( iEdg ` B ) o. ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) ) ` i ) ) |
156 |
|
f1of |
|- ( g : E -1-1-onto-> K -> g : E --> K ) |
157 |
|
eqid |
|- E = E |
158 |
157 19
|
feq23i |
|- ( g : E --> K <-> g : E --> ran ( iEdg ` B ) ) |
159 |
156 158
|
sylib |
|- ( g : E -1-1-onto-> K -> g : E --> ran ( iEdg ` B ) ) |
160 |
159
|
adantr |
|- ( ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) -> g : E --> ran ( iEdg ` B ) ) |
161 |
|
f1of |
|- ( ( iEdg ` A ) : dom ( iEdg ` A ) -1-1-onto-> E -> ( iEdg ` A ) : dom ( iEdg ` A ) --> E ) |
162 |
32 161
|
syl |
|- ( A e. USHGraph -> ( iEdg ` A ) : dom ( iEdg ` A ) --> E ) |
163 |
162
|
ad2antrr |
|- ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) -> ( iEdg ` A ) : dom ( iEdg ` A ) --> E ) |
164 |
|
fco |
|- ( ( g : E --> ran ( iEdg ` B ) /\ ( iEdg ` A ) : dom ( iEdg ` A ) --> E ) -> ( g o. ( iEdg ` A ) ) : dom ( iEdg ` A ) --> ran ( iEdg ` B ) ) |
165 |
160 163 164
|
syl2anr |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) -> ( g o. ( iEdg ` A ) ) : dom ( iEdg ` A ) --> ran ( iEdg ` B ) ) |
166 |
165
|
anim1i |
|- ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) -> ( ( g o. ( iEdg ` A ) ) : dom ( iEdg ` A ) --> ran ( iEdg ` B ) /\ i e. dom ( iEdg ` A ) ) ) |
167 |
166
|
adantr |
|- ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) /\ ( ( iEdg ` A ) ` i ) e. ran ( iEdg ` A ) ) -> ( ( g o. ( iEdg ` A ) ) : dom ( iEdg ` A ) --> ran ( iEdg ` B ) /\ i e. dom ( iEdg ` A ) ) ) |
168 |
|
ffvelrn |
|- ( ( ( g o. ( iEdg ` A ) ) : dom ( iEdg ` A ) --> ran ( iEdg ` B ) /\ i e. dom ( iEdg ` A ) ) -> ( ( g o. ( iEdg ` A ) ) ` i ) e. ran ( iEdg ` B ) ) |
169 |
167 168
|
syl |
|- ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) /\ ( ( iEdg ` A ) ` i ) e. ran ( iEdg ` A ) ) -> ( ( g o. ( iEdg ` A ) ) ` i ) e. ran ( iEdg ` B ) ) |
170 |
|
fvresi |
|- ( ( ( g o. ( iEdg ` A ) ) ` i ) e. ran ( iEdg ` B ) -> ( ( _I |` ran ( iEdg ` B ) ) ` ( ( g o. ( iEdg ` A ) ) ` i ) ) = ( ( g o. ( iEdg ` A ) ) ` i ) ) |
171 |
169 170
|
syl |
|- ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) /\ ( ( iEdg ` A ) ` i ) e. ran ( iEdg ` A ) ) -> ( ( _I |` ran ( iEdg ` B ) ) ` ( ( g o. ( iEdg ` A ) ) ` i ) ) = ( ( g o. ( iEdg ` A ) ) ` i ) ) |
172 |
162
|
ad3antrrr |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) -> ( iEdg ` A ) : dom ( iEdg ` A ) --> E ) |
173 |
172
|
anim1i |
|- ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) -> ( ( iEdg ` A ) : dom ( iEdg ` A ) --> E /\ i e. dom ( iEdg ` A ) ) ) |
174 |
173
|
adantr |
|- ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) /\ ( ( iEdg ` A ) ` i ) e. ran ( iEdg ` A ) ) -> ( ( iEdg ` A ) : dom ( iEdg ` A ) --> E /\ i e. dom ( iEdg ` A ) ) ) |
175 |
|
fvco3 |
|- ( ( ( iEdg ` A ) : dom ( iEdg ` A ) --> E /\ i e. dom ( iEdg ` A ) ) -> ( ( g o. ( iEdg ` A ) ) ` i ) = ( g ` ( ( iEdg ` A ) ` i ) ) ) |
176 |
174 175
|
syl |
|- ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) /\ ( ( iEdg ` A ) ` i ) e. ran ( iEdg ` A ) ) -> ( ( g o. ( iEdg ` A ) ) ` i ) = ( g ` ( ( iEdg ` A ) ` i ) ) ) |
177 |
171 176
|
eqtrd |
|- ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) /\ ( ( iEdg ` A ) ` i ) e. ran ( iEdg ` A ) ) -> ( ( _I |` ran ( iEdg ` B ) ) ` ( ( g o. ( iEdg ` A ) ) ` i ) ) = ( g ` ( ( iEdg ` A ) ` i ) ) ) |
178 |
143 155 177
|
3eqtr3rd |
|- ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) /\ ( ( iEdg ` A ) ` i ) e. ran ( iEdg ` A ) ) -> ( g ` ( ( iEdg ` A ) ` i ) ) = ( ( ( iEdg ` B ) o. ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) ) ` i ) ) |
179 |
|
dff1o4 |
|- ( ( iEdg ` B ) : dom ( iEdg ` B ) -1-1-onto-> K <-> ( ( iEdg ` B ) Fn dom ( iEdg ` B ) /\ `' ( iEdg ` B ) Fn K ) ) |
180 |
22 179
|
sylib |
|- ( B e. USHGraph -> ( ( iEdg ` B ) Fn dom ( iEdg ` B ) /\ `' ( iEdg ` B ) Fn K ) ) |
181 |
180
|
simprd |
|- ( B e. USHGraph -> `' ( iEdg ` B ) Fn K ) |
182 |
181
|
ad5antlr |
|- ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) /\ ( ( iEdg ` A ) ` i ) e. ran ( iEdg ` A ) ) -> `' ( iEdg ` B ) Fn K ) |
183 |
156
|
adantr |
|- ( ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) -> g : E --> K ) |
184 |
134 183
|
anim12ci |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) -> ( g : E --> K /\ ( iEdg ` A ) : dom ( iEdg ` A ) --> E ) ) |
185 |
184
|
ad2antrr |
|- ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) /\ ( ( iEdg ` A ) ` i ) e. ran ( iEdg ` A ) ) -> ( g : E --> K /\ ( iEdg ` A ) : dom ( iEdg ` A ) --> E ) ) |
186 |
|
fco |
|- ( ( g : E --> K /\ ( iEdg ` A ) : dom ( iEdg ` A ) --> E ) -> ( g o. ( iEdg ` A ) ) : dom ( iEdg ` A ) --> K ) |
187 |
185 186
|
syl |
|- ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) /\ ( ( iEdg ` A ) ` i ) e. ran ( iEdg ` A ) ) -> ( g o. ( iEdg ` A ) ) : dom ( iEdg ` A ) --> K ) |
188 |
|
fnfco |
|- ( ( `' ( iEdg ` B ) Fn K /\ ( g o. ( iEdg ` A ) ) : dom ( iEdg ` A ) --> K ) -> ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) Fn dom ( iEdg ` A ) ) |
189 |
182 187 188
|
syl2anc |
|- ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) /\ ( ( iEdg ` A ) ` i ) e. ran ( iEdg ` A ) ) -> ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) Fn dom ( iEdg ` A ) ) |
190 |
|
fvco2 |
|- ( ( ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) Fn dom ( iEdg ` A ) /\ i e. dom ( iEdg ` A ) ) -> ( ( ( iEdg ` B ) o. ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) ) ` i ) = ( ( iEdg ` B ) ` ( ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) ` i ) ) ) |
191 |
189 141 190
|
syl2anc |
|- ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) /\ ( ( iEdg ` A ) ` i ) e. ran ( iEdg ` A ) ) -> ( ( ( iEdg ` B ) o. ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) ) ` i ) = ( ( iEdg ` B ) ` ( ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) ` i ) ) ) |
192 |
130 178 191
|
3eqtrd |
|- ( ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) /\ ( ( iEdg ` A ) ` i ) e. ran ( iEdg ` A ) ) -> ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) ` i ) ) ) |
193 |
121 192
|
mpdan |
|- ( ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) /\ i e. dom ( iEdg ` A ) ) -> ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) ` i ) ) ) |
194 |
193
|
ralrimiva |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) -> A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) ` i ) ) ) |
195 |
116 194
|
jca |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) -> ( ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) ` i ) ) ) ) |
196 |
|
f1oeq1 |
|- ( h = ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) -> ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) <-> ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) ) ) |
197 |
|
fveq1 |
|- ( h = ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) -> ( h ` i ) = ( ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) ` i ) ) |
198 |
197
|
fveq2d |
|- ( h = ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) -> ( ( iEdg ` B ) ` ( h ` i ) ) = ( ( iEdg ` B ) ` ( ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) ` i ) ) ) |
199 |
198
|
eqeq2d |
|- ( h = ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) -> ( ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) <-> ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) ` i ) ) ) ) |
200 |
199
|
ralbidv |
|- ( h = ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) -> ( A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) <-> A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) ` i ) ) ) ) |
201 |
196 200
|
anbi12d |
|- ( h = ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) -> ( ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) <-> ( ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( ( `' ( iEdg ` B ) o. ( g o. ( iEdg ` A ) ) ) ` i ) ) ) ) ) |
202 |
104 195 201
|
spcedv |
|- ( ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) /\ ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) -> E. h ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) |
203 |
202
|
ex |
|- ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) -> ( ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) -> E. h ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) ) |
204 |
203
|
exlimdv |
|- ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) -> ( E. g ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) -> E. h ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) ) |
205 |
99 204
|
impbid |
|- ( ( ( A e. USHGraph /\ B e. USHGraph ) /\ f : V -1-1-onto-> W ) -> ( E. h ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) <-> E. g ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) ) |
206 |
205
|
pm5.32da |
|- ( ( A e. USHGraph /\ B e. USHGraph ) -> ( ( f : V -1-1-onto-> W /\ E. h ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) <-> ( f : V -1-1-onto-> W /\ E. g ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) ) ) |
207 |
206
|
exbidv |
|- ( ( A e. USHGraph /\ B e. USHGraph ) -> ( E. f ( f : V -1-1-onto-> W /\ E. h ( h : dom ( iEdg ` A ) -1-1-onto-> dom ( iEdg ` B ) /\ A. i e. dom ( iEdg ` A ) ( f " ( ( iEdg ` A ) ` i ) ) = ( ( iEdg ` B ) ` ( h ` i ) ) ) ) <-> E. f ( f : V -1-1-onto-> W /\ E. g ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) ) ) |
208 |
7 207
|
bitrd |
|- ( ( A e. USHGraph /\ B e. USHGraph ) -> ( A IsomGr B <-> E. f ( f : V -1-1-onto-> W /\ E. g ( g : E -1-1-onto-> K /\ A. e e. E ( f " e ) = ( g ` e ) ) ) ) ) |