Theorem isausgren 15922

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 3630	. . . . 5 |- ( v = V -> ~P v = ~P V )
3	2	adantr 276	. . . 4 |- ( ( v = V /\ e = E ) -> ~P v = ~P V )
4	3	rabeqdv 2771	. . 3 |- ( ( v = V /\ e = E ) -> { x e. ~P v | x ~~ 2o } = { x e. ~P V | x ~~ 2o } )
5	1, 4	sseq12d 3233	. 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 4329	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 1373   e. wcel 2178   {crab 2490   C_ wss 3175   ~Pcpw 3627   class class class wbr 4060   {copab 4121   2oc2o 6521   ~~ cen 6850

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4179  ax-pow 4235  ax-pr 4270

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rab 2495  df-v 2779  df-un 3179  df-in 3181  df-ss 3188  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-br 4061  df-opab 4123

This theorem is referenced by:  ausgrusgrben  15923  usgrausgrien  15924  ausgrumgrien  15925  ausgrusgrien  15926