Theorem edg0usgr 26145

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		edgval 25941	. . . . 5 |- (Edg ` G ) = ran (iEdg ` G )
2	1	a1i 11	. . . 4 |- ( G e. W -> (Edg ` G ) = ran (iEdg ` G ) )
3	2	eqeq1d 2624	. . 3 |- ( G e. W -> ( (Edg ` G ) = (/) <-> ran (iEdg ` G ) = (/) ) )
4		funrel 5905	. . . . . 6 |- ( Fun (iEdg ` G ) -> Rel (iEdg ` G ) )
5		relrn0 5383	. . . . . . 7 |- ( Rel (iEdg ` G ) -> ( (iEdg ` G ) = (/) <-> ran (iEdg ` G ) = (/) ) )
6	5	bicomd 213	. . . . . 6 |- ( Rel (iEdg ` G ) -> ( ran (iEdg ` G ) = (/) <-> (iEdg ` G ) = (/) ) )
7	4, 6	syl 17	. . . . 5 |- ( Fun (iEdg ` G ) -> ( ran (iEdg ` G ) = (/) <-> (iEdg ` G ) = (/) ) )
8		simpr 477	. . . . . . 7 |- ( ( (iEdg ` G ) = (/) /\ G e. W ) -> G e. W )
9		simpl 473	. . . . . . 7 |- ( ( (iEdg ` G ) = (/) /\ G e. W ) -> (iEdg ` G ) = (/) )
10	8, 9	usgr0e 26128	. . . . . 6 |- ( ( (iEdg ` G ) = (/) /\ G e. W ) -> G e. USGraph )
11	10	ex 450	. . . . 5 |- ( (iEdg ` G ) = (/) -> ( G e. W -> G e. USGraph ) )
12	7, 11	syl6bi 243	. . . 4 |- ( Fun (iEdg ` G ) -> ( ran (iEdg ` G ) = (/) -> ( G e. W -> G e. USGraph ) ) )
13	12	com13 88	. . 3 |- ( G e. W -> ( ran (iEdg ` G ) = (/) -> ( Fun (iEdg ` G ) -> G e. USGraph ) ) )
14	3, 13	sylbid 230	. 2 |- ( G e. W -> ( (Edg ` G ) = (/) -> ( Fun (iEdg ` G ) -> G e. USGraph ) ) )
15	14	3imp 1256	1 |- ( ( G e. W /\ (Edg ` G ) = (/) /\ Fun (iEdg ` G ) ) -> G e. USGraph )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   (/)c0 3915   ran crn 5115   Rel wrel 5119   Fun wfun 5882   ` cfv 5888  iEdgciedg 25875  Edgcedg 25939   USGraph cusgr 26044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fv 5896  df-edg 25940  df-usgr 26046

This theorem is referenced by: (None)