Theorem isuspgrop 26056

Description: The property of being an undirected simple pseudograph represented as an ordered pair. The representation as an ordered pair is the usual representation of a graph, see section I.1 of [Bollobas] p. 1. (Contributed by AV, 25-Nov-2021.)

Assertion

Ref	Expression
isuspgrop	|- ( ( V e. W /\ E e. X ) -> ( <. V , E >. e. USPGraph <-> E : dom E -1-1-> { p e. ( ~P V \ { (/) } ) | ( # ` p ) <_ 2 } ) )

Distinct variable groups:   E, p   V, p   W, p   X, p

Proof of Theorem isuspgrop

Step	Hyp	Ref	Expression
1		opex 4932	. . 3 |- <. V , E >. e. _V
2		eqid 2622	. . . 4 |- (Vtx ` <. V , E >. ) = (Vtx ` <. V , E >. )
3		eqid 2622	. . . 4 |- (iEdg ` <. V , E >. ) = (iEdg ` <. V , E >. )
4	2, 3	isuspgr 26047	. . 3 |- ( <. V , E >. e. _V -> ( <. V , E >. e. USPGraph <-> (iEdg ` <. V , E >. ) : dom (iEdg ` <. V , E >. ) -1-1-> { p e. ( ~P (Vtx ` <. V , E >. ) \ { (/) } ) | ( # ` p ) <_ 2 } ) )
5	1, 4	mp1i 13	. 2 |- ( ( V e. W /\ E e. X ) -> ( <. V , E >. e. USPGraph <-> (iEdg ` <. V , E >. ) : dom (iEdg ` <. V , E >. ) -1-1-> { p e. ( ~P (Vtx ` <. V , E >. ) \ { (/) } ) | ( # ` p ) <_ 2 } ) )
6		opiedgfv 25887	. . 3 |- ( ( V e. W /\ E e. X ) -> (iEdg ` <. V , E >. ) = E )
7	6	dmeqd 5326	. . 3 |- ( ( V e. W /\ E e. X ) -> dom (iEdg ` <. V , E >. ) = dom E )
8		opvtxfv 25884	. . . . . 6 |- ( ( V e. W /\ E e. X ) -> (Vtx ` <. V , E >. ) = V )
9	8	pweqd 4163	. . . . 5 |- ( ( V e. W /\ E e. X ) -> ~P (Vtx ` <. V , E >. ) = ~P V )
10	9	difeq1d 3727	. . . 4 |- ( ( V e. W /\ E e. X ) -> ( ~P (Vtx ` <. V , E >. ) \ { (/) } ) = ( ~P V \ { (/) } ) )
11	10	rabeqdv 3194	. . 3 |- ( ( V e. W /\ E e. X ) -> { p e. ( ~P (Vtx ` <. V , E >. ) \ { (/) } ) | ( # ` p ) <_ 2 } = { p e. ( ~P V \ { (/) } ) | ( # ` p ) <_ 2 } )
12	6, 7, 11	f1eq123d 6131	. 2 |- ( ( V e. W /\ E e. X ) -> ( (iEdg ` <. V , E >. ) : dom (iEdg ` <. V , E >. ) -1-1-> { p e. ( ~P (Vtx ` <. V , E >. ) \ { (/) } ) | ( # ` p ) <_ 2 } <-> E : dom E -1-1-> { p e. ( ~P V \ { (/) } ) | ( # ` p ) <_ 2 } ) )
13	5, 12	bitrd 268	1 |- ( ( V e. W /\ E e. X ) -> ( <. V , E >. e. USPGraph <-> E : dom E -1-1-> { p e. ( ~P V \ { (/) } ) | ( # ` p ) <_ 2 } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   {crab 2916   _Vcvv 3200   \ cdif 3571   (/)c0 3915   ~Pcpw 4158   {csn 4177   <.cop 4183   class class class wbr 4653   dom cdm 5114   -1-1->wf1 5885   ` cfv 5888   <_ cle 10075   2c2 11070   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875   USPGraph cuspgr 26043

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fv 5896  df-1st 7168  df-2nd 7169  df-vtx 25876  df-iedg 25877  df-uspgr 26045

This theorem is referenced by:  uspgrsprfo  41756