Theorem edg0iedg0g 15916

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 15909	. . . . 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 4960	. . . 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 5343	. . . 4 |- ( Fun I -> Rel I )
12		relrn0 4994	. . . . 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 1397   e. wcel 2202   (/)c0 3494   ran crn 4726   Rel wrel 4730   Fun wfun 5320   ` cfv 5326  iEdgciedg 15863  Edgcedg 15907

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-2nd 6303  df-sub 8351  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-9 9208  df-n0 9402  df-dec 9611  df-ndx 13084  df-slot 13085  df-edgf 15855  df-iedg 15865  df-edg 15908

This theorem is referenced by:  uhgriedg0edg0  15985  egrsubgr  16113