Theorem edg0iedg0g 16307

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 16300	. . . . 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 5010	. . . 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 5394	. . . 4 |- ( Fun I -> Rel I )
12		relrn0 5044	. . . . 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 = (/) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   (/)c0 3520   ran crn 4775   Rel wrel 4779   Fun wfun 5371   ` cfv 5377  iEdgciedg 16254  Edgcedg 16298

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-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-2nd 6375  df-sub 8499  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-dec 9778  df-ndx 13355  df-slot 13356  df-edgf 16246  df-iedg 16256  df-edg 16299

This theorem is used by:  uhgriedg0edg0  16376  egrsubgr  16504