Theorem isausgra 23415

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 3963	. . . . 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 3065	. . 3 |- ( ( v = V /\ e = E ) -> { x e. ~P v | ( # ` x ) = 2 } = { x e. ~P V | ( # ` x ) = 2 } )
6	1, 5	sseq12d 3485	. 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 4703	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 1370   e. wcel 1758   {crab 2799   C_ wss 3428   ~Pcpw 3960   class class class wbr 4392   {copab 4449   ` cfv 5518   2c2 10474   #chash 12206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-br 4393  df-opab 4451

This theorem is referenced by:  ausisusgra  23416