Theorem isausgren 16149

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 3671	. . . . 5 |- ( v = V -> ~P v = ~P V )
3	2	adantr 276	. . . 4 |- ( ( v = V /\ e = E ) -> ~P v = ~P V )
4	3	rabeqdv 2806	. . 3 |- ( ( v = V /\ e = E ) -> { x e. ~P v | x ~~ 2o } = { x e. ~P V | x ~~ 2o } )
5	1, 4	sseq12d 3268	. 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 4381	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 1398   e. wcel 2203   {crab 2524   C_ wss 3210   ~Pcpw 3668   class class class wbr 4108   {copab 4169   2oc2o 6640   ~~ cen 6972

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rab 2529  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109  df-opab 4171

This theorem is referenced by:  ausgrusgrben  16150  usgrausgrien  16151  ausgrumgrien  16152  ausgrusgrien  16153