Theorem umgrislfupgrdom 15970

Description: A multigraph is a loop-free pseudograph. (Contributed by AV, 27-Jan-2021.)

Hypotheses

Ref	Expression
umgrislfupgr.v	|- V = (Vtx ` G )
umgrislfupgr.i	|- I = (iEdg ` G )

Assertion

Ref	Expression
umgrislfupgrdom	|- ( G e. UMGraph <-> ( G e. UPGraph /\ I : dom I --> { x e. ~P V | 2o ~<_ x } ) )

Distinct variable groups:   x, G   x, V

Allowed substitution hint:   I( x)

Proof of Theorem umgrislfupgrdom

Step	Hyp	Ref	Expression
1		umgrupgr 15953	. . 3 |- ( G e. UMGraph -> G e. UPGraph )
2		umgrislfupgr.v	. . . . 5 |- V = (Vtx ` G )
3		umgrislfupgr.i	. . . . 5 |- I = (iEdg ` G )
4	2, 3	umgrfen 15948	. . . 4 |- ( G e. UMGraph -> I : dom I --> { x e. ~P V | x ~~ 2o } )
5		id 19	. . . . 5 |- ( I : dom I --> { x e. ~P V | x ~~ 2o } -> I : dom I --> { x e. ~P V | x ~~ 2o } )
6		ensymb 6949	. . . . . . . . 9 |- ( 2o ~~ x <-> x ~~ 2o )
7		endom 6931	. . . . . . . . 9 |- ( 2o ~~ x -> 2o ~<_ x )
8	6, 7	sylbir 135	. . . . . . . 8 |- ( x ~~ 2o -> 2o ~<_ x )
9	8	a1i 9	. . . . . . 7 |- ( x e. ~P V -> ( x ~~ 2o -> 2o ~<_ x ) )
10	9	ss2rabi 3307	. . . . . 6 |- { x e. ~P V | x ~~ 2o } C_ { x e. ~P V | 2o ~<_ x }
11	10	a1i 9	. . . . 5 |- ( I : dom I --> { x e. ~P V | x ~~ 2o } -> { x e. ~P V | x ~~ 2o } C_ { x e. ~P V | 2o ~<_ x } )
12	5, 11	fssd 5492	. . . 4 |- ( I : dom I --> { x e. ~P V | x ~~ 2o } -> I : dom I --> { x e. ~P V | 2o ~<_ x } )
13	4, 12	syl 14	. . 3 |- ( G e. UMGraph -> I : dom I --> { x e. ~P V | 2o ~<_ x } )
14	1, 13	jca 306	. 2 |- ( G e. UMGraph -> ( G e. UPGraph /\ I : dom I --> { x e. ~P V | 2o ~<_ x } ) )
15	2, 3	upgrfen 15938	. . . 4 |- ( G e. UPGraph -> I : dom I --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
16		fin 5520	. . . . 5 |- ( I : dom I --> ( { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } i^i { x e. ~P V | 2o ~<_ x } ) <-> ( I : dom I --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } /\ I : dom I --> { x e. ~P V | 2o ~<_ x } ) )
17		umgrislfupgrenlem 15969	. . . . . 6 |- ( { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } i^i { x e. ~P V | 2o ~<_ x } ) = { x e. ~P V | x ~~ 2o }
18		feq3 5464	. . . . . 6 |- ( ( { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } i^i { x e. ~P V | 2o ~<_ x } ) = { x e. ~P V | x ~~ 2o } -> ( I : dom I --> ( { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } i^i { x e. ~P V | 2o ~<_ x } ) <-> I : dom I --> { x e. ~P V | x ~~ 2o } ) )
19	17, 18	ax-mp 5	. . . . 5 |- ( I : dom I --> ( { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } i^i { x e. ~P V | 2o ~<_ x } ) <-> I : dom I --> { x e. ~P V | x ~~ 2o } )
20	16, 19	sylbb1 137	. . . 4 |- ( ( I : dom I --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } /\ I : dom I --> { x e. ~P V | 2o ~<_ x } ) -> I : dom I --> { x e. ~P V | x ~~ 2o } )
21	15, 20	sylan 283	. . 3 |- ( ( G e. UPGraph /\ I : dom I --> { x e. ~P V | 2o ~<_ x } ) -> I : dom I --> { x e. ~P V | x ~~ 2o } )
22	2, 3	isumgren 15946	. . . 4 |- ( G e. UPGraph -> ( G e. UMGraph <-> I : dom I --> { x e. ~P V | x ~~ 2o } ) )
23	22	adantr 276	. . 3 |- ( ( G e. UPGraph /\ I : dom I --> { x e. ~P V | 2o ~<_ x } ) -> ( G e. UMGraph <-> I : dom I --> { x e. ~P V | x ~~ 2o } ) )
24	21, 23	mpbird 167	. 2 |- ( ( G e. UPGraph /\ I : dom I --> { x e. ~P V | 2o ~<_ x } ) -> G e. UMGraph )
25	14, 24	impbii 126	1 |- ( G e. UMGraph <-> ( G e. UPGraph /\ I : dom I --> { x e. ~P V | 2o ~<_ x } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   = wceq 1395   e. wcel 2200   {crab 2512   i^i cin 3197   C_ wss 3198   ~Pcpw 3650   class class class wbr 4086   dom cdm 4723   -->wf 5320   ` cfv 5324   1oc1o 6570   2oc2o 6571   ~~ cen 6902   ~<_ cdom 6903  Vtxcvtx 15853  iEdgciedg 15854  UPGraphcupgr 15932  UMGraphcumgr 15933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-1o 6577  df-2o 6578  df-er 6697  df-en 6905  df-dom 6906  df-sub 8342  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-5 9195  df-6 9196  df-7 9197  df-8 9198  df-9 9199  df-n0 9393  df-dec 9602  df-ndx 13075  df-slot 13076  df-base 13078  df-edgf 15846  df-vtx 15855  df-iedg 15856  df-upgren 15934  df-umgren 15935

This theorem is referenced by:  vtxdumgrfival  16104