Theorem uhgr0vb 15724

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 2206	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
2		eqid 2206	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
3	1, 2	uhgrfm 15713	. . 3 |- ( G e. UHGraph -> (iEdg ` G ) : dom (iEdg ` G ) --> { s e. ~P (Vtx ` G ) | E. j j e. s } )
4		pweq 3620	. . . . . . . 8 |- ( (Vtx ` G ) = (/) -> ~P (Vtx ` G ) = ~P (/) )
5	4	rabeqdv 2767	. . . . . . 7 |- ( (Vtx ` G ) = (/) -> { s e. ~P (Vtx ` G ) | E. j j e. s } = { s e. ~P (/) | E. j j e. s } )
6		pw0ss 15723	. . . . . . 7 |- { s e. ~P (/) | E. j j e. s } = (/)
7	5, 6	eqtrdi 2255	. . . . . 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 5420	. . . 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 5474	. . . . 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 15722	. . . 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 1373   E.wex 1516   e. wcel 2177   {crab 2489   (/)c0 3461   ~Pcpw 3617   dom cdm 4679   -->wf 5272   ` cfv 5276  Vtxcvtx 15655  iEdgciedg 15656  UHGraphcuhgr 15707

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-cnre 8043

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-fo 5282  df-fv 5284  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-sub 8252  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-5 9105  df-6 9106  df-7 9107  df-8 9108  df-9 9109  df-n0 9303  df-dec 9512  df-ndx 12879  df-slot 12880  df-base 12882  df-edgf 15648  df-vtx 15657  df-iedg 15658  df-uhgrm 15709

This theorem is referenced by: (None)