Theorem isausgra 24017

Description: The property of an unordered pair to be an alternatively defined undirected simple graph without loops (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
ausgra.1	|- G = { <. v , e >. | e C_ { x e. ~P v | ( # ` x ) = 2 } }

Assertion

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

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 isausgra

Step	Hyp	Ref	Expression
1		simpr 461	. . 3 |- ( ( v = V /\ e = E ) -> e = E )
2		pweq 4006	. . . . 5 |- ( v = V -> ~P v = ~P V )
3	2	adantr 465	. . . 4 |- ( ( v = V /\ e = E ) -> ~P v = ~P V )
4		biidd 237	. . . 4 |- ( ( v = V /\ e = E ) -> ( ( # ` x ) = 2 <-> ( # ` x ) = 2 ) )
5	3, 4	rabeqbidv 3101	. . 3 |- ( ( v = V /\ e = E ) -> { x e. ~P v | ( # ` x ) = 2 } = { x e. ~P V | ( # ` x ) = 2 } )
6	1, 5	sseq12d 3526	. 2 |- ( ( v = V /\ e = E ) -> ( e C_ { x e. ~P v | ( # ` x ) = 2 } <-> E C_ { x e. ~P V | ( # ` x ) = 2 } ) )
7		ausgra.1	. 2 |- G = { <. v , e >. | e C_ { x e. ~P v | ( # ` x ) = 2 } }
8	6, 7	brabga 4754	1 |- ( ( V e. W /\ E e. X ) -> ( V G E <-> E C_ { x e. ~P V | ( # ` x ) = 2 } ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1374   e. wcel 1762   {crab 2811   C_ wss 3469   ~Pcpw 4003   class class class wbr 4440   {copab 4497   ` cfv 5579   2c2 10574   #chash 12360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499

This theorem is referenced by:  ausisusgra  24018