Theorem usgrislfuspgrdom 16172

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

Hypotheses

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

Assertion

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

Distinct variable groups:   x, G   x, V

Allowed substitution hint:   I( x)

Proof of Theorem usgrislfuspgrdom

Step	Hyp	Ref	Expression
1		usgruspgr 16165	. . 3 |- ( G e. USGraph -> G e. USPGraph )
2		usgrislfuspgr.v	. . . . 5 |- V = (Vtx ` G )
3		usgrislfuspgr.i	. . . . 5 |- I = (iEdg ` G )
4	2, 3	usgrfen 16142	. . . 4 |- ( G e. USGraph -> I : dom I -1-1-> { x e. ~P V | x ~~ 2o } )
5		f1f 5572	. . . . 5 |- ( I : dom I -1-1-> { x e. ~P V | x ~~ 2o } -> I : dom I --> { x e. ~P V | x ~~ 2o } )
6		ensym 7020	. . . . . . . . 9 |- ( x ~~ 2o -> 2o ~~ x )
7		endom 7001	. . . . . . . . 9 |- ( 2o ~~ x -> 2o ~<_ x )
8	6, 7	syl 14	. . . . . . . 8 |- ( x ~~ 2o -> 2o ~<_ x )
9	8	a1i 9	. . . . . . 7 |- ( x e. ~P V -> ( x ~~ 2o -> 2o ~<_ x ) )
10	9	ss2rabi 3319	. . . . . 6 |- { x e. ~P V | x ~~ 2o } C_ { x e. ~P V | 2o ~<_ x }
11	10	a1i 9	. . . . 5 |- ( I : dom I -1-1-> { x e. ~P V | x ~~ 2o } -> { x e. ~P V | x ~~ 2o } C_ { x e. ~P V | 2o ~<_ x } )
12	5, 11	fssd 5521	. . . 4 |- ( I : dom I -1-1-> { x e. ~P V | x ~~ 2o } -> I : dom I --> { x e. ~P V | 2o ~<_ x } )
13	4, 12	syl 14	. . 3 |- ( G e. USGraph -> I : dom I --> { x e. ~P V | 2o ~<_ x } )
14	1, 13	jca 306	. 2 |- ( G e. USGraph -> ( G e. USPGraph /\ I : dom I --> { x e. ~P V | 2o ~<_ x } ) )
15	2, 3	uspgrfen 16141	. . . 4 |- ( G e. USPGraph -> I : dom I -1-1-> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
16		df-f1 5356	. . . . . 6 |- ( I : dom I -1-1-> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } <-> ( I : dom I --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } /\ Fun `' I ) )
17		fin 5552	. . . . . . . . . . 11 |- ( 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 } ) )
18		umgrislfupgrenlem 16112	. . . . . . . . . . . 12 |- ( { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } i^i { x e. ~P V | 2o ~<_ x } ) = { x e. ~P V | x ~~ 2o }
19		feq3 5492	. . . . . . . . . . . 12 |- ( ( { 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 } ) )
20	18, 19	ax-mp 5	. . . . . . . . . . 11 |- ( 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 } )
21	17, 20	sylbb1 137	. . . . . . . . . 10 |- ( ( 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 } )
22	21	anim1i 340	. . . . . . . . 9 |- ( ( ( I : dom I --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } /\ I : dom I --> { x e. ~P V | 2o ~<_ x } ) /\ Fun `' I ) -> ( I : dom I --> { x e. ~P V | x ~~ 2o } /\ Fun `' I ) )
23		df-f1 5356	. . . . . . . . 9 |- ( I : dom I -1-1-> { x e. ~P V | x ~~ 2o } <-> ( I : dom I --> { x e. ~P V | x ~~ 2o } /\ Fun `' I ) )
24	22, 23	sylibr 134	. . . . . . . 8 |- ( ( ( I : dom I --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } /\ I : dom I --> { x e. ~P V | 2o ~<_ x } ) /\ Fun `' I ) -> I : dom I -1-1-> { x e. ~P V | x ~~ 2o } )
25	24	ex 115	. . . . . . 7 |- ( ( I : dom I --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } /\ I : dom I --> { x e. ~P V | 2o ~<_ x } ) -> ( Fun `' I -> I : dom I -1-1-> { x e. ~P V | x ~~ 2o } ) )
26	25	impancom 260	. . . . . 6 |- ( ( I : dom I --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } /\ Fun `' I ) -> ( I : dom I --> { x e. ~P V | 2o ~<_ x } -> I : dom I -1-1-> { x e. ~P V | x ~~ 2o } ) )
27	16, 26	sylbi 121	. . . . 5 |- ( I : dom I -1-1-> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } -> ( I : dom I --> { x e. ~P V | 2o ~<_ x } -> I : dom I -1-1-> { x e. ~P V | x ~~ 2o } ) )
28	27	imp 124	. . . 4 |- ( ( I : dom I -1-1-> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } /\ I : dom I --> { x e. ~P V | 2o ~<_ x } ) -> I : dom I -1-1-> { x e. ~P V | x ~~ 2o } )
29	15, 28	sylan 283	. . 3 |- ( ( G e. USPGraph /\ I : dom I --> { x e. ~P V | 2o ~<_ x } ) -> I : dom I -1-1-> { x e. ~P V | x ~~ 2o } )
30	2, 3	isusgren 16140	. . . 4 |- ( G e. USPGraph -> ( G e. USGraph <-> I : dom I -1-1-> { x e. ~P V | x ~~ 2o } ) )
31	30	adantr 276	. . 3 |- ( ( G e. USPGraph /\ I : dom I --> { x e. ~P V | 2o ~<_ x } ) -> ( G e. USGraph <-> I : dom I -1-1-> { x e. ~P V | x ~~ 2o } ) )
32	29, 31	mpbird 167	. 2 |- ( ( G e. USPGraph /\ I : dom I --> { x e. ~P V | 2o ~<_ x } ) -> G e. USGraph )
33	14, 32	impbii 126	1 |- ( G e. USGraph <-> ( G e. USPGraph /\ 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 3209   C_ wss 3210   ~Pcpw 3668   class class class wbr 4108   `'ccnv 4747   dom cdm 4748   Fun wfun 5345   -->wf 5347   -1-1->wf1 5348   ` cfv 5351   1oc1o 6639   2oc2o 6640   ~~ cen 6972   ~<_ cdom 6973  Vtxcvtx 15994  iEdgciedg 15995  USPGraphcuspgr 16135  USGraphcusgr 16136

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 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-cnre 8234

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 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-1o 6646  df-2o 6647  df-er 6766  df-en 6975  df-dom 6976  df-sub 8442  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-5 9295  df-6 9296  df-7 9297  df-8 9298  df-9 9299  df-n0 9493  df-dec 9706  df-ndx 13204  df-slot 13205  df-base 13207  df-edgf 15987  df-vtx 15996  df-iedg 15997  df-uspgren 16137  df-usgren 16138

This theorem is referenced by: (None)