Theorem wrdupgr

Description: The property of being an undirected pseudograph, expressing the edges as "words". (Contributed by Mario Carneiro, 11-Mar-2015) (Revised by AV, 10-Oct-2020)

	Ref	Expression
Hypotheses	isupgr.v	|- V = ( Vtx ` G )
	isupgr.e	|- E = ( iEdg ` G )
Assertion	wrdupgr	|- ( ( G e. U /\ E e. Word X ) -> ( G e. UPGraph <-> E e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } ) )

Proof

Step	Hyp	Ref	Expression
1		isupgr.v	|- V = ( Vtx ` G )
2		isupgr.e	|- E = ( iEdg ` G )
3	1 2	isupgr	|- ( G e. U -> ( G e. UPGraph <-> E : dom E --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } ) )
4	3	adantr	|- ( ( G e. U /\ E e. Word X ) -> ( G e. UPGraph <-> E : dom E --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } ) )
5		wrdf	|- ( E e. Word X -> E : ( 0 ..^ ( # ` E ) ) --> X )
6	5	adantl	|- ( ( G e. U /\ E e. Word X ) -> E : ( 0 ..^ ( # ` E ) ) --> X )
7	6	fdmd	|- ( ( G e. U /\ E e. Word X ) -> dom E = ( 0 ..^ ( # ` E ) ) )
8	7	feq2d	|- ( ( G e. U /\ E e. Word X ) -> ( E : dom E --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } <-> E : ( 0 ..^ ( # ` E ) ) --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } ) )
9		iswrdi	|- ( E : ( 0 ..^ ( # ` E ) ) --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } -> E e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
10		wrdf	|- ( E e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } -> E : ( 0 ..^ ( # ` E ) ) --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
11	9 10	impbii	|- ( E : ( 0 ..^ ( # ` E ) ) --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } <-> E e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
12	8 11	bitrdi	|- ( ( G e. U /\ E e. Word X ) -> ( E : dom E --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } <-> E e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } ) )
13	4 12	bitrd	|- ( ( G e. U /\ E e. Word X ) -> ( G e. UPGraph <-> E e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } ) )