Theorem uhgr0vb 16337

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 2238	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
2		eqid 2238	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
3	1, 2	uhgrfm 16326	. . 3 |- ( G e. UHGraph -> (iEdg ` G ) : dom (iEdg ` G ) --> { s e. ~P (Vtx ` G ) | E. j j e. s } )
4		pweq 3691	. . . . . . . 8 |- ( (Vtx ` G ) = (/) -> ~P (Vtx ` G ) = ~P (/) )
5	4	rabeqdv 2815	. . . . . . 7 |- ( (Vtx ` G ) = (/) -> { s e. ~P (Vtx ` G ) | E. j j e. s } = { s e. ~P (/) | E. j j e. s } )
6		pw0ss 16336	. . . . . . 7 |- { s e. ~P (/) | E. j j e. s } = (/)
7	5, 6	eqtrdi 2287	. . . . . 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 5522	. . . 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 5584	. . . . 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 16335	. . . 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 ) = (/) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   {crab 2532   (/)c0 3520   ~Pcpw 3688   dom cdm 4774   -->wf 5373   ` cfv 5377  Vtxcvtx 16265  iEdgciedg 16266  UHGraphcuhgr 16320

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fo 5383  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-sub 8499  df-inn 9306  df-2 9364  df-3 9365  df-4 9366  df-5 9367  df-6 9368  df-7 9369  df-8 9370  df-9 9371  df-n0 9566  df-dec 9780  df-ndx 13357  df-slot 13358  df-base 13360  df-edgf 16258  df-vtx 16267  df-iedg 16268  df-uhgrm 16322

This theorem is used by:  usgr0vb  16486  uhgr0v0e  16487  0uhgrsubgr  16518