Theorem edg0usgr 16091

Description: A class without edges is a simple graph. Since ran F = (/) does not generally imply Fun F, but Fun (iEdg ` G ) is required for G to be a simple graph, however, this must be provided as assertion. (Contributed by AV, 18-Oct-2020.)

Assertion

Ref	Expression
edg0usgr	|- ( ( G e. W /\ (Edg ` G ) = (/) /\ Fun (iEdg ` G ) ) -> G e. USGraph )

Proof of Theorem edg0usgr

Step	Hyp	Ref	Expression
1		edgvalg 15903	. . . 4 |- ( G e. W -> (Edg ` G ) = ran (iEdg ` G ) )
2	1	eqeq1d 2238	. . 3 |- ( G e. W -> ( (Edg ` G ) = (/) <-> ran (iEdg ` G ) = (/) ) )
3		funrel 5341	. . . . . 6 |- ( Fun (iEdg ` G ) -> Rel (iEdg ` G ) )
4		relrn0 4992	. . . . . . 7 |- ( Rel (iEdg ` G ) -> ( (iEdg ` G ) = (/) <-> ran (iEdg ` G ) = (/) ) )
5	4	bicomd 141	. . . . . 6 |- ( Rel (iEdg ` G ) -> ( ran (iEdg ` G ) = (/) <-> (iEdg ` G ) = (/) ) )
6	3, 5	syl 14	. . . . 5 |- ( Fun (iEdg ` G ) -> ( ran (iEdg ` G ) = (/) <-> (iEdg ` G ) = (/) ) )
7		simpr 110	. . . . . . 7 |- ( ( (iEdg ` G ) = (/) /\ G e. W ) -> G e. W )
8		simpl 109	. . . . . . 7 |- ( ( (iEdg ` G ) = (/) /\ G e. W ) -> (iEdg ` G ) = (/) )
9	7, 8	usgr0e 16076	. . . . . 6 |- ( ( (iEdg ` G ) = (/) /\ G e. W ) -> G e. USGraph )
10	9	ex 115	. . . . 5 |- ( (iEdg ` G ) = (/) -> ( G e. W -> G e. USGraph ) )
11	6, 10	biimtrdi 163	. . . 4 |- ( Fun (iEdg ` G ) -> ( ran (iEdg ` G ) = (/) -> ( G e. W -> G e. USGraph ) ) )
12	11	com13 80	. . 3 |- ( G e. W -> ( ran (iEdg ` G ) = (/) -> ( Fun (iEdg ` G ) -> G e. USGraph ) ) )
13	2, 12	sylbid 150	. 2 |- ( G e. W -> ( (Edg ` G ) = (/) -> ( Fun (iEdg ` G ) -> G e. USGraph ) ) )
14	13	3imp 1217	1 |- ( ( G e. W /\ (Edg ` G ) = (/) /\ Fun (iEdg ` G ) ) -> G e. USGraph )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   (/)c0 3492   ran crn 4724   Rel wrel 4728   Fun wfun 5318   ` cfv 5324  iEdgciedg 15857  Edgcedg 15901  USGraphcusgr 15998

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-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-cnre 8136

This theorem depends on definitions:  df-bi 117  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-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-sub 8345  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-5 9198  df-6 9199  df-7 9200  df-8 9201  df-9 9202  df-n0 9396  df-dec 9605  df-ndx 13078  df-slot 13079  df-base 13081  df-edgf 15849  df-vtx 15858  df-iedg 15859  df-edg 15902  df-usgren 16000

This theorem is referenced by: (None)