Theorem edg0iedg0g 16221

Description: There is no edge in a graph iff its edge function is empty. (Contributed by AV, 15-Dec-2020.) (Revised by AV, 8-Dec-2021.)

Hypotheses

Ref	Expression
edg0iedg0.i	|- I = (iEdg ` G )
edg0iedg0.e	|- E = (Edg ` G )

Assertion

Ref	Expression
edg0iedg0g	|- ( ( G e. V /\ Fun I ) -> ( E = (/) <-> I = (/) ) )

Proof of Theorem edg0iedg0g

Step	Hyp	Ref	Expression
1		edg0iedg0.e	. . . . 5 |- E = (Edg ` G )
2		edgvalg 16214	. . . . 5 |- ( G e. V -> (Edg ` G ) = ran (iEdg ` G ) )
3	1, 2	eqtrid 2283	. . . 4 |- ( G e. V -> E = ran (iEdg ` G ) )
4	3	eqeq1d 2247	. . 3 |- ( G e. V -> ( E = (/) <-> ran (iEdg ` G ) = (/) ) )
5	4	adantr 276	. 2 |- ( ( G e. V /\ Fun I ) -> ( E = (/) <-> ran (iEdg ` G ) = (/) ) )
6		edg0iedg0.i	. . . . . 6 |- I = (iEdg ` G )
7	6	eqcomi 2242	. . . . 5 |- (iEdg ` G ) = I
8	7	rneqi 5005	. . . 4 |- ran (iEdg ` G ) = ran I
9	8	eqeq1i 2246	. . 3 |- ( ran (iEdg ` G ) = (/) <-> ran I = (/) )
10	9	a1i 9	. 2 |- ( ( G e. V /\ Fun I ) -> ( ran (iEdg ` G ) = (/) <-> ran I = (/) ) )
11		funrel 5389	. . . 4 |- ( Fun I -> Rel I )
12		relrn0 5039	. . . . 5 |- ( Rel I -> ( I = (/) <-> ran I = (/) ) )
13	12	bicomd 141	. . . 4 |- ( Rel I -> ( ran I = (/) <-> I = (/) ) )
14	11, 13	syl 14	. . 3 |- ( Fun I -> ( ran I = (/) <-> I = (/) ) )
15	14	adantl 277	. 2 |- ( ( G e. V /\ Fun I ) -> ( ran I = (/) <-> I = (/) ) )
16	5, 10, 15	3bitrd 214	1 |- ( ( G e. V /\ Fun I ) -> ( E = (/) <-> I = (/) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   (/)c0 3520   ran crn 4770   Rel wrel 4774   Fun wfun 5366   ` cfv 5372  iEdgciedg 16168  Edgcedg 16212

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 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 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fo 5378  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-2nd 6365  df-sub 8489  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-dec 9757  df-ndx 13333  df-slot 13334  df-edgf 16160  df-iedg 16170  df-edg 16213

This theorem is referenced by:  uhgriedg0edg0  16290  egrsubgr  16418