Theorem uhgr0vb 25967

Description: The null graph, with no vertices, is a hypergraph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 9-Oct-2020.)

Assertion

Ref	Expression
uhgr0vb	|- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> ( G e. UHGraph <-> (iEdg ` G ) = (/) ) )

Proof of Theorem uhgr0vb

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
2		eqid 2622	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
3	1, 2	uhgrf 25957	. . 3 |- ( G e. UHGraph -> (iEdg ` G ) : dom (iEdg ` G ) --> ( ~P (Vtx ` G ) \ { (/) } ) )
4		pweq 4161	. . . . . . . 8 |- ( (Vtx ` G ) = (/) -> ~P (Vtx ` G ) = ~P (/) )
5	4	difeq1d 3727	. . . . . . 7 |- ( (Vtx ` G ) = (/) -> ( ~P (Vtx ` G ) \ { (/) } ) = ( ~P (/) \ { (/) } ) )
6		pw0 4343	. . . . . . . . 9 |- ~P (/) = { (/) }
7	6	difeq1i 3724	. . . . . . . 8 |- ( ~P (/) \ { (/) } ) = ( { (/) } \ { (/) } )
8		difid 3948	. . . . . . . 8 |- ( { (/) } \ { (/) } ) = (/)
9	7, 8	eqtri 2644	. . . . . . 7 |- ( ~P (/) \ { (/) } ) = (/)
10	5, 9	syl6eq 2672	. . . . . 6 |- ( (Vtx ` G ) = (/) -> ( ~P (Vtx ` G ) \ { (/) } ) = (/) )
11	10	adantl 482	. . . . 5 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> ( ~P (Vtx ` G ) \ { (/) } ) = (/) )
12	11	feq3d 6032	. . . 4 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> ( (iEdg ` G ) : dom (iEdg ` G ) --> ( ~P (Vtx ` G ) \ { (/) } ) <-> (iEdg ` G ) : dom (iEdg ` G ) --> (/) ) )
13		f00 6087	. . . . 5 |- ( (iEdg ` G ) : dom (iEdg ` G ) --> (/) <-> ( (iEdg ` G ) = (/) /\ dom (iEdg ` G ) = (/) ) )
14	13	simplbi 476	. . . 4 |- ( (iEdg ` G ) : dom (iEdg ` G ) --> (/) -> (iEdg ` G ) = (/) )
15	12, 14	syl6bi 243	. . 3 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> ( (iEdg ` G ) : dom (iEdg ` G ) --> ( ~P (Vtx ` G ) \ { (/) } ) -> (iEdg ` G ) = (/) ) )
16	3, 15	syl5 34	. 2 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> ( G e. UHGraph -> (iEdg ` G ) = (/) ) )
17		simpl 473	. . . . 5 |- ( ( G e. W /\ (iEdg ` G ) = (/) ) -> G e. W )
18		simpr 477	. . . . 5 |- ( ( G e. W /\ (iEdg ` G ) = (/) ) -> (iEdg ` G ) = (/) )
19	17, 18	uhgr0e 25966	. . . 4 |- ( ( G e. W /\ (iEdg ` G ) = (/) ) -> G e. UHGraph )
20	19	ex 450	. . 3 |- ( G e. W -> ( (iEdg ` G ) = (/) -> G e. UHGraph ) )
21	20	adantr 481	. 2 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> ( (iEdg ` G ) = (/) -> G e. UHGraph ) )
22	16, 21	impbid 202	1 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> ( G e. UHGraph <-> (iEdg ` G ) = (/) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571   (/)c0 3915   ~Pcpw 4158   {csn 4177   dom cdm 5114   -->wf 5884   ` cfv 5888  Vtxcvtx 25874  iEdgciedg 25875   UHGraph cuhgr 25951

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-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-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-fv 5896  df-uhgr 25953

This theorem is referenced by:  usgr0vb  26129  uhgr0v0e  26130  0uhgrsubgr  26171  finsumvtxdg2size  26446  0uhgrrusgr  26474  frgr0v  27125  frgruhgr0v  27127