Step |
Hyp |
Ref |
Expression |
1 |
|
subumgredg2.v |
|- V = ( Vtx ` S ) |
2 |
|
subumgredg2.i |
|- I = ( iEdg ` S ) |
3 |
|
fveqeq2 |
|- ( e = ( I ` X ) -> ( ( # ` e ) = 2 <-> ( # ` ( I ` X ) ) = 2 ) ) |
4 |
|
umgruhgr |
|- ( G e. UMGraph -> G e. UHGraph ) |
5 |
4
|
3ad2ant2 |
|- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> G e. UHGraph ) |
6 |
|
simp1 |
|- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> S SubGraph G ) |
7 |
|
simp3 |
|- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> X e. dom I ) |
8 |
1 2 5 6 7
|
subgruhgredgd |
|- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> ( I ` X ) e. ( ~P V \ { (/) } ) ) |
9 |
|
eqid |
|- ( iEdg ` G ) = ( iEdg ` G ) |
10 |
9
|
uhgrfun |
|- ( G e. UHGraph -> Fun ( iEdg ` G ) ) |
11 |
4 10
|
syl |
|- ( G e. UMGraph -> Fun ( iEdg ` G ) ) |
12 |
11
|
3ad2ant2 |
|- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> Fun ( iEdg ` G ) ) |
13 |
|
eqid |
|- ( Vtx ` S ) = ( Vtx ` S ) |
14 |
|
eqid |
|- ( Vtx ` G ) = ( Vtx ` G ) |
15 |
|
eqid |
|- ( Edg ` S ) = ( Edg ` S ) |
16 |
13 14 2 9 15
|
subgrprop2 |
|- ( S SubGraph G -> ( ( Vtx ` S ) C_ ( Vtx ` G ) /\ I C_ ( iEdg ` G ) /\ ( Edg ` S ) C_ ~P ( Vtx ` S ) ) ) |
17 |
16
|
simp2d |
|- ( S SubGraph G -> I C_ ( iEdg ` G ) ) |
18 |
17
|
3ad2ant1 |
|- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> I C_ ( iEdg ` G ) ) |
19 |
|
funssfv |
|- ( ( Fun ( iEdg ` G ) /\ I C_ ( iEdg ` G ) /\ X e. dom I ) -> ( ( iEdg ` G ) ` X ) = ( I ` X ) ) |
20 |
19
|
eqcomd |
|- ( ( Fun ( iEdg ` G ) /\ I C_ ( iEdg ` G ) /\ X e. dom I ) -> ( I ` X ) = ( ( iEdg ` G ) ` X ) ) |
21 |
12 18 7 20
|
syl3anc |
|- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> ( I ` X ) = ( ( iEdg ` G ) ` X ) ) |
22 |
21
|
fveq2d |
|- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> ( # ` ( I ` X ) ) = ( # ` ( ( iEdg ` G ) ` X ) ) ) |
23 |
|
simp2 |
|- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> G e. UMGraph ) |
24 |
2
|
dmeqi |
|- dom I = dom ( iEdg ` S ) |
25 |
24
|
eleq2i |
|- ( X e. dom I <-> X e. dom ( iEdg ` S ) ) |
26 |
|
subgreldmiedg |
|- ( ( S SubGraph G /\ X e. dom ( iEdg ` S ) ) -> X e. dom ( iEdg ` G ) ) |
27 |
26
|
ex |
|- ( S SubGraph G -> ( X e. dom ( iEdg ` S ) -> X e. dom ( iEdg ` G ) ) ) |
28 |
25 27
|
syl5bi |
|- ( S SubGraph G -> ( X e. dom I -> X e. dom ( iEdg ` G ) ) ) |
29 |
28
|
a1d |
|- ( S SubGraph G -> ( G e. UMGraph -> ( X e. dom I -> X e. dom ( iEdg ` G ) ) ) ) |
30 |
29
|
3imp |
|- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> X e. dom ( iEdg ` G ) ) |
31 |
14 9
|
umgredg2 |
|- ( ( G e. UMGraph /\ X e. dom ( iEdg ` G ) ) -> ( # ` ( ( iEdg ` G ) ` X ) ) = 2 ) |
32 |
23 30 31
|
syl2anc |
|- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> ( # ` ( ( iEdg ` G ) ` X ) ) = 2 ) |
33 |
22 32
|
eqtrd |
|- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> ( # ` ( I ` X ) ) = 2 ) |
34 |
3 8 33
|
elrabd |
|- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> ( I ` X ) e. { e e. ( ~P V \ { (/) } ) | ( # ` e ) = 2 } ) |
35 |
|
prprrab |
|- { e e. ( ~P V \ { (/) } ) | ( # ` e ) = 2 } = { e e. ~P V | ( # ` e ) = 2 } |
36 |
34 35
|
eleqtrdi |
|- ( ( S SubGraph G /\ G e. UMGraph /\ X e. dom I ) -> ( I ` X ) e. { e e. ~P V | ( # ` e ) = 2 } ) |