Theorem isausgren 15965

Description: The property of an ordered pair to be an alternatively defined simple graph, defined as a pair (V,E) of a set V (vertex set) and a set of unordered pairs of elements of V (edge set). (Contributed by Alexander van der Vekens, 28-Aug-2017.)

Hypothesis

Ref	Expression
ausgr.1	|- G = { <. v , e >. | e C_ { x e. ~P v | x ~~ 2o } }

Assertion

Ref	Expression
isausgren	|- ( ( V e. W /\ E e. X ) -> ( V G E <-> E C_ { x e. ~P V | x ~~ 2o } ) )

Distinct variable groups:   v, e, x, E   e, V, v, x   x, X

Allowed substitution hints:   G( x, v, e)   W( x, v, e)   X( v, e)

Proof of Theorem isausgren

Step	Hyp	Ref	Expression
1		simpr 110	. . 3 |- ( ( v = V /\ e = E ) -> e = E )
2		pweq 3652	. . . . 5 |- ( v = V -> ~P v = ~P V )
3	2	adantr 276	. . . 4 |- ( ( v = V /\ e = E ) -> ~P v = ~P V )
4	3	rabeqdv 2793	. . 3 |- ( ( v = V /\ e = E ) -> { x e. ~P v | x ~~ 2o } = { x e. ~P V | x ~~ 2o } )
5	1, 4	sseq12d 3255	. 2 |- ( ( v = V /\ e = E ) -> ( e C_ { x e. ~P v | x ~~ 2o } <-> E C_ { x e. ~P V | x ~~ 2o } ) )
6		ausgr.1	. 2 |- G = { <. v , e >. | e C_ { x e. ~P v | x ~~ 2o } }
7	5, 6	brabga 4352	1 |- ( ( V e. W /\ E e. X ) -> ( V G E <-> E C_ { x e. ~P V | x ~~ 2o } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   {crab 2512   C_ wss 3197   ~Pcpw 3649   class class class wbr 4083   {copab 4144   2oc2o 6556   ~~ cen 6885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146

This theorem is referenced by:  ausgrusgrben  15966  usgrausgrien  15967  ausgrumgrien  15968  ausgrusgrien  15969