Theorem uspgrushgr 26070

Description: A simple pseudograph is an undirected simple hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 15-Oct-2020.)

Assertion

Ref	Expression
uspgrushgr	|- ( G e. USPGraph -> G e. USHGraph )

Proof of Theorem uspgrushgr

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 |- (Vtx ` G ) = (Vtx ` G )
2		eqid 2622	. . . . 5 |- (iEdg ` G ) = (iEdg ` G )
3	1, 2	isuspgr 26047	. . . 4 |- ( G e. USPGraph -> ( G e. USPGraph <-> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } ) )
4		ssrab2 3687	. . . . 5 |- { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } C_ ( ~P (Vtx ` G ) \ { (/) } )
5		f1ss 6106	. . . . 5 |- ( ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } /\ { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } C_ ( ~P (Vtx ` G ) \ { (/) } ) ) -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> ( ~P (Vtx ` G ) \ { (/) } ) )
6	4, 5	mpan2 707	. . . 4 |- ( (iEdg ` G ) : dom (iEdg ` G ) -1-1-> { x e. ( ~P (Vtx ` G ) \ { (/) } ) | ( # ` x ) <_ 2 } -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> ( ~P (Vtx ` G ) \ { (/) } ) )
7	3, 6	syl6bi 243	. . 3 |- ( G e. USPGraph -> ( G e. USPGraph -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> ( ~P (Vtx ` G ) \ { (/) } ) ) )
8	1, 2	isushgr 25956	. . 3 |- ( G e. USPGraph -> ( G e. USHGraph <-> (iEdg ` G ) : dom (iEdg ` G ) -1-1-> ( ~P (Vtx ` G ) \ { (/) } ) ) )
9	7, 8	sylibrd 249	. 2 |- ( G e. USPGraph -> ( G e. USPGraph -> G e. USHGraph ) )
10	9	pm2.43i 52	1 |- ( G e. USPGraph -> G e. USHGraph )

Syntax hints:   -> wi 4   e. wcel 1990   {crab 2916   \ cdif 3571   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   class class class wbr 4653   dom cdm 5114   -1-1->wf1 5885   ` cfv 5888   <_ cle 10075   2c2 11070   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875   USHGraph cushgr 25952   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

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-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-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-ushgr 25954  df-uspgr 26045

This theorem is referenced by:  uspgrupgrushgr  26072  usgredgedg  26122  vtxdusgrfvedg  26387  1loopgrvd2  26399