Theorem edg0iedg0g 15777

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 15771	. . . . 5 |- ( G e. V -> (Edg ` G ) = ran (iEdg ` G ) )
3	1, 2	eqtrid 2252	. . . 4 |- ( G e. V -> E = ran (iEdg ` G ) )
4	3	eqeq1d 2216	. . 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 2211	. . . . 5 |- (iEdg ` G ) = I
8	7	rneqi 4925	. . . 4 |- ran (iEdg ` G ) = ran I
9	8	eqeq1i 2215	. . 3 |- ( ran (iEdg ` G ) = (/) <-> ran I = (/) )
10	9	a1i 9	. 2 |- ( ( G e. V /\ Fun I ) -> ( ran (iEdg ` G ) = (/) <-> ran I = (/) ) )
11		funrel 5307	. . . 4 |- ( Fun I -> Rel I )
12		relrn0 4959	. . . . 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 1373   e. wcel 2178   (/)c0 3468   ran crn 4694   Rel wrel 4698   Fun wfun 5284   ` cfv 5290  iEdgciedg 15727  Edgcedg 15769

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 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071

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 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fo 5296  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-2nd 6250  df-sub 8280  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-5 9133  df-6 9134  df-7 9135  df-8 9136  df-9 9137  df-n0 9331  df-dec 9540  df-ndx 12950  df-slot 12951  df-edgf 15719  df-iedg 15729  df-edg 15770

This theorem is referenced by:  uhgriedg0edg0  15841