Theorem umgrislfupgrdom 15923

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 15906	. . 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 15901	. . . 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 6930	. . . . . . . . 9 |- ( 2o ~~ x <-> x ~~ 2o )
7		endom 6912	. . . . . . . . 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 3306	. . . . . 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 5485	. . . 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 15891	. . . 4 |- ( G e. UPGraph -> I : dom I --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
16		fin 5511	. . . . 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 15922	. . . . . 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 5457	. . . . . 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 15899	. . . 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 3196   C_ wss 3197   ~Pcpw 3649   class class class wbr 4082   dom cdm 4718   -->wf 5313   ` cfv 5317   1oc1o 6553   2oc2o 6554   ~~ cen 6883   ~<_ cdom 6884  Vtxcvtx 15807  iEdgciedg 15808  UPGraphcupgr 15885  UMGraphcumgr 15886

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 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-1o 6560  df-2o 6561  df-er 6678  df-en 6886  df-dom 6887  df-sub 8315  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-5 9168  df-6 9169  df-7 9170  df-8 9171  df-9 9172  df-n0 9366  df-dec 9575  df-ndx 13030  df-slot 13031  df-base 13033  df-edgf 15800  df-vtx 15809  df-iedg 15810  df-upgren 15887  df-umgren 15888

This theorem is referenced by: (None)