Theorem umgrislfupgrdom 16126

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 16107	. . 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 16102	. . . 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 7020	. . . . . . . . 9 |- ( 2o ~~ x <-> x ~~ 2o )
7		endom 7002	. . . . . . . . 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 3320	. . . . . 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 5522	. . . 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 16092	. . . 4 |- ( G e. UPGraph -> I : dom I --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
16		fin 5553	. . . . 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 16125	. . . . . 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 5493	. . . . . 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 16100	. . . 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 716   = wceq 1398   e. wcel 2203   {crab 2524   i^i cin 3210   C_ wss 3211   ~Pcpw 3669   class class class wbr 4109   dom cdm 4749   -->wf 5348   ` cfv 5352   1oc1o 6640   2oc2o 6641   ~~ cen 6973   ~<_ cdom 6974  Vtxcvtx 16007  iEdgciedg 16008  UPGraphcupgr 16086  UMGraphcumgr 16087

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-1o 6647  df-2o 6648  df-er 6767  df-en 6976  df-dom 6977  df-sub 8446  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-9 9303  df-n0 9497  df-dec 9710  df-ndx 13215  df-slot 13216  df-base 13218  df-edgf 16000  df-vtx 16009  df-iedg 16010  df-upgren 16088  df-umgren 16089

This theorem is referenced by:  vtxdumgrfival  16293