Theorem isausgr 26059

Description: The property of an unordered 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 ) = 2 } }

Assertion

Ref	Expression
isausgr	|- ( ( 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 isausgr

Step	Hyp	Ref	Expression
1		simpr 477	. . 3 |- ( ( v = V /\ e = E ) -> e = E )
2		pweq 4161	. . . . 5 |- ( v = V -> ~P v = ~P V )
3	2	adantr 481	. . . 4 |- ( ( v = V /\ e = E ) -> ~P v = ~P V )
4	3	rabeqdv 3194	. . 3 |- ( ( v = V /\ e = E ) -> { x e. ~P v | ( # ` x ) = 2 } = { x e. ~P V | ( # ` x ) = 2 } )
5	1, 4	sseq12d 3634	. 2 |- ( ( v = V /\ e = E ) -> ( e C_ { x e. ~P v | ( # ` x ) = 2 } <-> E C_ { x e. ~P V | ( # ` x ) = 2 } ) )
6		ausgr.1	. 2 |- G = { <. v , e >. | e C_ { x e. ~P v | ( # ` x ) = 2 } }
7	5, 6	brabga 4989	1 |- ( ( V e. W /\ E e. X ) -> ( V G E <-> E C_ { x e. ~P V | ( # ` x ) = 2 } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   {copab 4712   ` cfv 5888   2c2 11070   #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713

This theorem is referenced by:  ausgrusgrb  26060  usgrausgri  26061  ausgrumgri  26062  ausgrusgri  26063