Theorem isausgren 16322

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 3688	. . . . 5 |- ( v = V -> ~P v = ~P V )
3	2	adantr 276	. . . 4 |- ( ( v = V /\ e = E ) -> ~P v = ~P V )
4	3	rabeqdv 2815	. . 3 |- ( ( v = V /\ e = E ) -> { x e. ~P v | x ~~ 2o } = { x e. ~P V | x ~~ 2o } )
5	1, 4	sseq12d 3279	. 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 4401	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 1402   e. wcel 2209   {crab 2532   C_ wss 3220   ~Pcpw 3685   class class class wbr 4125   {copab 4186   2oc2o 6671   ~~ cen 7010

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188

This theorem is referenced by:  ausgrusgrben  16323  usgrausgrien  16324  ausgrumgrien  16325  ausgrusgrien  16326