Theorem isausgra 24771

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 459	. . 3 |- ( ( v = V /\ e = E ) -> e = E )
2		pweq 3958	. . . . 5 |- ( v = V -> ~P v = ~P V )
3	2	adantr 463	. . . 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 3054	. . 3 |- ( ( v = V /\ e = E ) -> { x e. ~P v | ( # ` x ) = 2 } = { x e. ~P V | ( # ` x ) = 2 } )
6	1, 5	sseq12d 3471	. 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 4704	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 367   = wceq 1405   e. wcel 1842   {crab 2758   C_ wss 3414   ~Pcpw 3955   class class class wbr 4395   {copab 4452   ` cfv 5569   2c2 10626   #chash 12452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454

This theorem is referenced by:  ausisusgra  24772