Step |
Hyp |
Ref |
Expression |
1 |
|
umgrislfupgr.v |
|- V = ( Vtx ` G ) |
2 |
|
umgrislfupgr.i |
|- I = ( iEdg ` G ) |
3 |
|
umgrupgr |
|- ( G e. UMGraph -> G e. UPGraph ) |
4 |
1 2
|
umgrf |
|- ( G e. UMGraph -> I : dom I --> { x e. ~P V | ( # ` x ) = 2 } ) |
5 |
|
id |
|- ( I : dom I --> { x e. ~P V | ( # ` x ) = 2 } -> I : dom I --> { x e. ~P V | ( # ` x ) = 2 } ) |
6 |
|
2re |
|- 2 e. RR |
7 |
6
|
leidi |
|- 2 <_ 2 |
8 |
7
|
a1i |
|- ( ( # ` x ) = 2 -> 2 <_ 2 ) |
9 |
|
breq2 |
|- ( ( # ` x ) = 2 -> ( 2 <_ ( # ` x ) <-> 2 <_ 2 ) ) |
10 |
8 9
|
mpbird |
|- ( ( # ` x ) = 2 -> 2 <_ ( # ` x ) ) |
11 |
10
|
a1i |
|- ( x e. ~P V -> ( ( # ` x ) = 2 -> 2 <_ ( # ` x ) ) ) |
12 |
11
|
ss2rabi |
|- { x e. ~P V | ( # ` x ) = 2 } C_ { x e. ~P V | 2 <_ ( # ` x ) } |
13 |
12
|
a1i |
|- ( I : dom I --> { x e. ~P V | ( # ` x ) = 2 } -> { x e. ~P V | ( # ` x ) = 2 } C_ { x e. ~P V | 2 <_ ( # ` x ) } ) |
14 |
5 13
|
fssd |
|- ( I : dom I --> { x e. ~P V | ( # ` x ) = 2 } -> I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) |
15 |
4 14
|
syl |
|- ( G e. UMGraph -> I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) |
16 |
3 15
|
jca |
|- ( G e. UMGraph -> ( G e. UPGraph /\ I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) ) |
17 |
1 2
|
upgrf |
|- ( G e. UPGraph -> I : dom I --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } ) |
18 |
|
fin |
|- ( I : dom I --> ( { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } i^i { x e. ~P V | 2 <_ ( # ` x ) } ) <-> ( I : dom I --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } /\ I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) ) |
19 |
|
umgrislfupgrlem |
|- ( { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } i^i { x e. ~P V | 2 <_ ( # ` x ) } ) = { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } |
20 |
|
feq3 |
|- ( ( { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } i^i { x e. ~P V | 2 <_ ( # ` x ) } ) = { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } -> ( I : dom I --> ( { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } i^i { x e. ~P V | 2 <_ ( # ` x ) } ) <-> I : dom I --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } ) ) |
21 |
19 20
|
ax-mp |
|- ( I : dom I --> ( { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } i^i { x e. ~P V | 2 <_ ( # ` x ) } ) <-> I : dom I --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } ) |
22 |
18 21
|
sylbb1 |
|- ( ( I : dom I --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } /\ I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) -> I : dom I --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } ) |
23 |
17 22
|
sylan |
|- ( ( G e. UPGraph /\ I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) -> I : dom I --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } ) |
24 |
1 2
|
isumgr |
|- ( G e. UPGraph -> ( G e. UMGraph <-> I : dom I --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } ) ) |
25 |
24
|
adantr |
|- ( ( G e. UPGraph /\ I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) -> ( G e. UMGraph <-> I : dom I --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } ) ) |
26 |
23 25
|
mpbird |
|- ( ( G e. UPGraph /\ I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) -> G e. UMGraph ) |
27 |
16 26
|
impbii |
|- ( G e. UMGraph <-> ( G e. UPGraph /\ I : dom I --> { x e. ~P V | 2 <_ ( # ` x ) } ) ) |