Theorem edg0iedg0g 15990

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 15983	. . . . 5 |- ( G e. V -> (Edg ` G ) = ran (iEdg ` G ) )
3	1, 2	eqtrid 2276	. . . 4 |- ( G e. V -> E = ran (iEdg ` G ) )
4	3	eqeq1d 2240	. . 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 2235	. . . . 5 |- (iEdg ` G ) = I
8	7	rneqi 4966	. . . 4 |- ran (iEdg ` G ) = ran I
9	8	eqeq1i 2239	. . 3 |- ( ran (iEdg ` G ) = (/) <-> ran I = (/) )
10	9	a1i 9	. 2 |- ( ( G e. V /\ Fun I ) -> ( ran (iEdg ` G ) = (/) <-> ran I = (/) ) )
11		funrel 5350	. . . 4 |- ( Fun I -> Rel I )
12		relrn0 5000	. . . . 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 1398   e. wcel 2202   (/)c0 3496   ran crn 4732   Rel wrel 4736   Fun wfun 5327   ` cfv 5333  iEdgciedg 15937  Edgcedg 15981

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fo 5339  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-2nd 6313  df-sub 8394  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-7 9249  df-8 9250  df-9 9251  df-n0 9445  df-dec 9656  df-ndx 13148  df-slot 13149  df-edgf 15929  df-iedg 15939  df-edg 15982

This theorem is referenced by:  uhgriedg0edg0  16059  egrsubgr  16187