Theorem uhgr0vb 16208

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

Dummy variables s j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2234	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
2		eqid 2234	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
3	1, 2	uhgrfm 16197	. . 3 |- ( G e. UHGraph -> (iEdg ` G ) : dom (iEdg ` G ) --> { s e. ~P (Vtx ` G ) | E. j j e. s } )
4		pweq 3677	. . . . . . . 8 |- ( (Vtx ` G ) = (/) -> ~P (Vtx ` G ) = ~P (/) )
5	4	rabeqdv 2809	. . . . . . 7 |- ( (Vtx ` G ) = (/) -> { s e. ~P (Vtx ` G ) | E. j j e. s } = { s e. ~P (/) | E. j j e. s } )
6		pw0ss 16207	. . . . . . 7 |- { s e. ~P (/) | E. j j e. s } = (/)
7	5, 6	eqtrdi 2283	. . . . . 6 |- ( (Vtx ` G ) = (/) -> { s e. ~P (Vtx ` G ) | E. j j e. s } = (/) )
8	7	adantl 277	. . . . 5 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> { s e. ~P (Vtx ` G ) | E. j j e. s } = (/) )
9	8	feq3d 5502	. . . 4 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> ( (iEdg ` G ) : dom (iEdg ` G ) --> { s e. ~P (Vtx ` G ) | E. j j e. s } <-> (iEdg ` G ) : dom (iEdg ` G ) --> (/) ) )
10		f00 5564	. . . . 5 |- ( (iEdg ` G ) : dom (iEdg ` G ) --> (/) <-> ( (iEdg ` G ) = (/) /\ dom (iEdg ` G ) = (/) ) )
11	10	simplbi 274	. . . 4 |- ( (iEdg ` G ) : dom (iEdg ` G ) --> (/) -> (iEdg ` G ) = (/) )
12	9, 11	biimtrdi 163	. . 3 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> ( (iEdg ` G ) : dom (iEdg ` G ) --> { s e. ~P (Vtx ` G ) | E. j j e. s } -> (iEdg ` G ) = (/) ) )
13	3, 12	syl5 32	. 2 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> ( G e. UHGraph -> (iEdg ` G ) = (/) ) )
14		simpl 109	. . . . 5 |- ( ( G e. W /\ (iEdg ` G ) = (/) ) -> G e. W )
15		simpr 110	. . . . 5 |- ( ( G e. W /\ (iEdg ` G ) = (/) ) -> (iEdg ` G ) = (/) )
16	14, 15	uhgr0e 16206	. . . 4 |- ( ( G e. W /\ (iEdg ` G ) = (/) ) -> G e. UHGraph )
17	16	ex 115	. . 3 |- ( G e. W -> ( (iEdg ` G ) = (/) -> G e. UHGraph ) )
18	17	adantr 276	. 2 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> ( (iEdg ` G ) = (/) -> G e. UHGraph ) )
19	13, 18	impbid 129	1 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> ( G e. UHGraph <-> (iEdg ` G ) = (/) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2205   {crab 2526   (/)c0 3512   ~Pcpw 3674   dom cdm 4754   -->wf 5353   ` cfv 5357  Vtxcvtx 16136  iEdgciedg 16137  UHGraphcuhgr 16191

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fo 5363  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-sub 8463  df-inn 9258  df-2 9316  df-3 9317  df-4 9318  df-5 9319  df-6 9320  df-7 9321  df-8 9322  df-9 9323  df-n0 9517  df-dec 9731  df-ndx 13302  df-slot 13303  df-base 13305  df-edgf 16129  df-vtx 16138  df-iedg 16139  df-uhgrm 16193

This theorem is referenced by:  usgr0vb  16357  uhgr0v0e  16358  0uhgrsubgr  16389